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II 



ERRATA FOR CAJORI'S HISTORY OF MATHEMATICS. 

Pages zi to xiii, Nos. 47, 67, 84, 88, 97. For Gedftchnias rt:ad GedflchtnlBS. 

No. 60. For 1887 read 1877. 
No. 65. For fur read ftir. 
No. 76, second line. For der read fttr. 
No. 06. For Beudiconti read Rendiconti. 

Page 3, line 4 up. For accomplishment, sacrificed read accomplish- 
ment, is said to have sacrificed. 

Page 31, Hue 10. For compass rtad compasses. 

Page 36, line 2 up. For VII. read XIL 

Page 39, lines 1 to 6 up. Chasles^s or Simson's definition of a Porism 
is preferable to Proclus's, given in the text See Gow, pp. 217-221. 

Page 52, line 3. For About 155 d.c. read About 155 d.c, or, according 
to M. Cantor, about 100 b.c. 

Page 62, line 1. For geometries read geometers. 

Page 75, line 19. For /fi rend 71. 

Page 106, line 1. For Mohammed ben Musa Uovarezmi read Mohammed 
ben Musa Al Hovarezmi, or rt'ad Muhammed ibn Musa Alchwarizmi. 

Page 106, line 3. For 814 read 813. 

Page 100, line 11. For Uovarezmi read Al Hovarezmi, or read Alch- 
warizmi. 

Page 111, line 14. For Fahri des Al Karhi read Abu Bekr Mohammed 
ben Al Hasan Al Karhi. 

Page 130, line 2. For Casso read Cassio. 

Page 1:^, line 14. For small scries read large series. 

Page 142, line 18. For go rend got. 

Page 174, line 26. For Midorge read Mydorge. 

Page 178, line 5 ; also, pp. 288, 289. Fttr Brauchiou read Brianchon. 

Page 220, line 12. For circle : — read circle, previously discow*r^d hv 
James Gregory : — 

Page 226, line 5. For Thomas Craigo read John Craig. 

Page 233, line 6. For 1725 rend 1726. 

Page 241, line 19. For ir,;jl-l708 read 1651-1708. 

Page 285, line 3. For 17(J6-1845 read 1765-1842. 

Page 308, line 3. For rntersuchunffen read Entwicklungen. 

Page 3.30, line 7. For Moor read Moore. 

Page :i35, line 12. For Sfries was made by Gauss read series, excepting 
a test for convergence by Maclaurin (afterwards re-discovered by 
Cauchy), was made by Gauss. 

Alphabetical Index. For Tchimhausen rmd Tschirnliausen ; under 
"Todhunter," for I., XIII. read XL Insert the following new 
references: under *»I)c Moivre" add 242, 245; under **Kleraents'' 
(Euclid's) add 01 ; under ** Laplace" a<ld 257 ; under ** Moivre, de " 
add 240, 242; under *» Substitutions" add 329. 



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A HISTORY OF MATHEMATICS 



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A HISTORY OF 



MATHEMATICS 



BY 



PLORIAN CAJORI 



yOBXSBLT PBOTBSSOB OF APPUSD MATHEMATICS IK THE TULANB UNIYERSITT 

OF Louibiana; kow Professor of Putsios 
IK Colorado College 



" I am sore that no sabject loses more than mathematics 
by any attempt to dissociate it from its history." — J. W. L. 

QULSBHMM 



^' Ntto gork 
MACMILLAN AND 

AND LONDON 
1894 



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JUN 20 19' 7 



1 ilAHbfLURF.D TO 
iMMNTAno ki)\^iiXf. ULdAi^i 



COPTBIGBT, 1888, 

bt macmillan and CO. 



J. S. Gushing & Co. — Berwick & Smith. 
Boston, Mass., U.S.A. 






PREFACE. 



-•O^ 



Ak increased interest in the history of the exact sciences 
manifested in recent years by teachers everywhere, and the 
attention given to historical inquiry in the mathematical 
class-rooms and seminaries of our leading universities, cause 
me to believe that a brief general History of Mathematics 
will be found acceptable to teachers and students. 

The pages treating — necessarily in a very condensed 
form — of the progress made during the present century, 
are put forth with great diffidence, although I have spent 
much time in the effort to render them accurate and 
reasonably complete. Many valuable suggestions and criti- 
cisms on the chapter on ''Becent Times" have been made 
by Dr. E. W. Davis, of the University of Nebraska. The 
proof-sheets of this chapter have also been submitted to 
Dr. J. E. Davies and Professor C. A. Van Velzer, both of the 
University of Wisconsin ; to Dr. G. B. Halsted, of the 
University of Texas ; Professor L. M. Hoskins, of the Leland 
Stanford Jr. University; and Professor G. D. Olds, of Amherst 
College, — all of whom have afforded valuable assistance. 
I am specially indebted to Professor F. H. Loud, of Colorado 
College, who has read the proof-sheets throughout. To all 

the gentlemen above named, as well as to Dr. Carlo Veneziani 

v 



Yl PREFACE. 

of Salt Lake City, who read the first part of my work in 

manuscript, I desire to express my hearty thanks. But in 

acknowledging their kindness, I trust that I shall not seem 

to lay upon them any share in the responsibility for errors 

which I may have introduced in subsequent revision of the 

text. 

FLOBIAN CAJOBL 

Colorado Collkgx, December, 1893. 



TABLE OF CONTENTS. 



INTBODUCTION 1 

ANTIQUITY 6 

The Babtlohians 6 

The Eotptians 9 

The Greeks 16 

Oreek Geometry 16 

The Ionic School 17 

The School of Pythagoras 19 

The Sophist School 23 

The Platonic School 29 

The First Alexandrian School 84 

The Second Alexandrian School 64 

Greek Arithmetic 63 

The Romaxs 77 

MIDDLE AGES 84 

The Hindoos 84 

The Aeabs 100 

EUSOPE DURING THE MiDDLB AgES 117 

Introduction of Roman Mathematics 117 

Translation of Arabic Manuscripts 124 

The First Awakening and its Sequel 128 

MODERN EUROPE 138 

The Renaissance 139 

YiETA TO Descartes 156 

Descartes to Newton 183 

Newton to Eulbr 199 

vii 



Vm TABLB OF CONTENTS. 

Paob 
EuLER, Lagrange, and Laplace 246 

The Origin of Modem Geometry 285 

RECENT TIMES 291 

Synthetic Geometry 293 

Analytic Geometry _-. . 307 

Algebra 315 

Analysis 331 

Theory op Functions 347 

Theory op Numbers 362 

Applied Mathematics 373 

INDEX 405 



BOOKS OF REFERENCE. 



The following books, pamphlets, and articles have been used 
in the preparation of this histoiy. Eeference to any of them 
is made in the text by giving the respective number. Histories 
marked with a star are the only ones of which extensive use 
has been made. 



1. G&NTHSR, S. Ziele und BesultaU der netieren MathematischrhiS' 

torischen Forschung. Erlangen, 1876. 

2. Cajori, F. Tfie Teaching and History of Mathematics in the U. 8, 

Washington, 1890. 

3. * Cantor, Moritz. Vorlesungen Uber Oeschichte der Mathematik. 

Leipzig. Bd. I., 1880 ; Bd. II., 1892. 

4. Efpiito, J. Astronomisches aus Babylon. Unter MUvoirkung von 

P. J. R. Strassmaibr. Freiburg, 1889. 
6. Bretschneider, C. A. Die Chometrie und die Geometer vor Eukli- 
des, Leipzig, 1870. 

6. * (jk>w, James. A Short History of Greek Mathematics. Cambridge, 

1884. 

7. • Hamkel, Hermann. Zur Geschichte der Mathematik im Alterthum 

und Mittelalter. Leipzig, 1874. 

8. ♦ Allman, G. J. Greek Geometry from Thales to Euclid. Dublin, 

1889. 

9. De Morgan, A. "Euclides" in Smith's Dictionary of Greek and 

Roman Biography and Mythology. 

10. Hankel, Hermann. Theorie der Complexen Zahlensyste^ne. Leip- 

zig, 1867. 

11. Whewbll, William. History of the Inductive Sciences. 

12. Zecthbn, H. O. Die Lehre von den Kegelschnitten im Alterthum. 

Kopenhagen, 1886. 

ix 



X BOOKS OF KEFEBENCE. 

13. *Cha8lbs, M. Geschichte der Oeometrie, Aub dem Franzosischen 

tibertragen durch Dr. L. A. Sohncke. Halle, 1839. 

14. Marie, Maximilien. Histoire des Sciences Mathimatiquea et Phy- 

siques, Tome I.-XII. Paris, 1883-1888. 
16. CoMTE, A. Philosophy of Mathematics^ translated by W. M. Gil- 
lespie. 

16. Hankel, Hermann. Die Entmckelung der MathemaHk in den letz- 

ten Jahrhunderten. Ttibingen, 1884. 

17. Gx)nth£r, Sieomund und Windelband, W. Creschichte der antiken 

Naturvoissenschaft und Philosophic. Nordllngen, 1888. 

18. Arneth, a. Geschichte der reinen Mathematik. Stuttgart, 1862. 
10. Cantor, Moritz. Mathematische Beitrdge sSum Kulturlehen der 

Volker. Halle, 1863. 

20. Mattuiessbn, Ludwio. OrundzUge der Antiken und Modemen 

Algebra der Litteralen Gleichungen, Leipzig, 1878. 

21. Ohrtmann und MOller. Fortschritte der Mathematik. 

22. Peacock, George. Article ^* Arithmetic/* in The Enq/clopasdia of 

Pure Mathematics. London, 1847. 

23. Herschel, J. F. W. Article ** Mathematics,*' in Edinburgh Ency- 

clopcedia. 

24. SuTER, Heinrich. Geschichte der Mathematischen Wissenschaften, 

Zttrich, 1873-76. 
26. Quetelet, a. Sciences Mathematiques et Physiques chez les Beiges. 
Bruxelles, 1866. 

26. Playfair, John. Article ** Progress of the Mathematical and Phys- 

ical Sciences,** in Encycloposdia Britannica, 7th edition, con- 
tinued in the 8th edition by Sir John Leslie. 

27. De Morgan, A. Arithmetical Books from the Invention of Printing 

to the Present Time. 

28. Napier, Mark. Memoirs of John Napier of Merchiston. Edin- 

burgh, 1834. 

29. Halsted, G. B. "Note on the First English Euclid," American 

Journal of Mathematics^ Vol. IT., 1879. 

30. Madame Perier. The Life of Mr. Paschal. Translated into 

English by W. A., London, 1744. 
^31. MoNTucLA, J. F. Histoire des Mathematiques. Paris, 1802. 

32. DOhring E. Kritische Geschichte der allgemeinen Principien der 

Mechanik. Leipzig, 1887. 

33. Brewster, D. The Memoirs of Newton. Edinburgh, 1860. 

34. Ball, W. W. R. A Short Account of the History of Mathematics. 

London, 1888, 2nd edition, 1893. 
— 35. De Morgan, A. "On the Early History of Infinitesimals,** in the 
Philosophical Magazine, November, 1852. 



BOOKS OF BEFERENCB. XI 

S6. Bibliatheca MatJiematica, herausgegeben von Gustajt EirsBTBdM, 
Stockholm. 

37. G&MTHER, SiEGMUND. VermUchU UrUersuchungen zur QeschicJUe 

der mathenKUischen Wissenschitften. Leipzig, 1876. 

38. ^Gbrhardt, C. I. Oeschichte der McUhematik in Dentschland, 

Mtinchen, 1877. 

39. Gbrhardt, C. L Entdeckung der Differenzialrechnung durch Leib- 

niz, HaUe, 1848. 

40. GxRHARDT, K. I. ** Leibniz in London,*' in Sitzungsberichte der 

Kdniglich Preussischen Academie der Wissenschc^ften zu Berlin^ 
Febmar, }801. 

41. Db Morgan, A. Articles ** Fluxions *' and **Commercium Epistoli- 

cum,** in the Penny Cydopasdia. 

42. ♦ToDHDicTER, I. A History of the Mathematical Theory of Probabil- 

ity from the Time of Pascal to that of Laplace. Cambridge and 
London, 1866. 

43. ♦ToDHUKTER, L A History of the Theory of Elasticity and of the 

Strength of Materials. Edited and completed by Karl Pearson. 
Cambridge, 1886. 

44. ToDHUNTER, L ** Notc on the History of Certain FormulsB in Spher- 

ical Trigonometry,'* Philosophical Magazine, February, 1873. 
46. Die Easier Mathematiker, Daniel Bernoulli und Leonhard Euler. 
Basel, 1884. 

46. Keipf, R. Oeschichte der Unendlichen Reihen. Tttbingen, 1889. 

47. Waltershausen, W. Sartorius. Qauss^ zum Oeddchniss. Leip- 

zig, 1866. 

48. Baumgart, Oswald. Ueber das Quadratische Beciprocitdtsgesetz. 

Leipzig, 1886. 

49. Hathaway, A. S. ** Early History of the Potential,** Bulletin of 

the N. r. Mathematical Society, I. 3. 

60. Wolf, Rudolf. Oeschichte der Astronomic. MUnchen, 1887. 

61. Arago, D. F. J. ** Eulogy on Laplace.** Translated by B. Powell, 

Smithsonian Report, 1874. 

62. BEArMONT, M. £lie de. ** Memoir of Legendre.** Translated by 

C. A. Alexander, Smithsonian Report, 1867. 

63. Arago, D. F. J. ** Joseph Fourier.** Smithsonian Report, 

1871. 

64. Wiener, Christian. Lehrbuch der Darstellenden Oeometrie, Leip- 

zig, 1884. 
66. *LoRiA, GiNO. Die Hauptsdchlichsten Theorien der Oeometrie in 
ihrer flrUheren und heutigen Entwickelung, ins deutsche Ubertra- 
gen von Fritz Scuutte. Leipzig, 1888. 



XU BOOKS OF BEFEBENCB. 

56. Catlby, Abthub. Inaogoral Address before the British Association, 
1888. 

67. Spottiswoodb, Willlam. Inaugural Address before the British 

Association, 1878. 

68. 61BB8, J. WiLLARD. ** Multiple Algebra,** Proceedings of the 

American AssocUUion for the Advancement of Science^ 1886. 
60. Fink, Karl. GeschichU der Elementar-Mathematik. Tttbingen, 
1890. 

60. WiTTSTBiN, Armin. Zuf Qeschichtc des ^fa^fattf <cAen Problems, 

Ndrdlingen, 1878. 

61. Klbin, Felix. Vergleichende Betrachtungen iiber neuere geome- 

trische Forschungen, Erlangen, 1872. 

62. Forsyth, A. R. Theory of Functions of a Complex Variable. 

Cambridge, 1803. 
^ 63. Graham, R. H. Geometry of Position. London, 1801. 

64. Schmidt, Franz. **Aus dem Leben zweier ungarischer Mathe- 

matiker Johann und Wolfgang Bolyai von Bolya.*' GrunerVs 

Archiv, 48 : 2, 1868. 
66. Fayaro, Anton. ** Justus Bellavitis,*' Zeitschrift fur Mathematik 

und Physik, 26 : 6, 1881. 

66. Dronke, Ad. Julius PlUcker. Bonn, 1871. 

67. Bauer, Gustav. Gedachnissrede auf Otto Hesse. Mtinchen, 

1882. 

68. Alfred Clebsch. Versuch einer Darlegung und Wurdigung seiner 

vaissenschaftlichen Leistungen von einigen seiner Freunde. Leip- 
zig, 1873. 
60. Haas, August. Versuch einer Darstellung der Geschichte des 
KrUmmungsmasses. TUbingen, 1881. 

70. Fine, Henry B. The Number-System of Algebra. Boston and 

New York, 1800. 

71. SoHLEOBL, Victor. Hermann Grassmann, sein Leben und seine 

Werke. Leipzig, 1878. 

72. Zahn, W. v. " Einige Worte zum Andenken an Hermann Hankel," 

Mathemalische Anndleny VII. 4, 1874. 

73. MuiR, Thomas. A Treatise on Determinants. 1882. 

74. Salmon, Gborob. "Arthur Cayley," Nature^ 28:21, September, 

1883. 

76. Cayley, A. "James Joseph Sylvester," Nature, 30:10, January, 
1880. 

76. BuRKHARDT, Heinrich. "Die Anf&nge der Gruppentheorie und 
Paolo RufiGbii,** Zeitschrift der Mathematik und Physik, Supple- 
ment, 1802. 



BOOKS OF BEFEBENGE. XIU 

77. Stlvesteb, J. J. Inaugural Pregidential Address to the Mathe- 

matical and Physical Section of the British Association at Exeter. 
1869. 

78. Valson, C. a. La Vie et les travaux du Baron Cauchy, Tome I., 

II., Paris, 1868. 

79. Sachse, Abnold. Versuch einer Oeschichte der Darstellung fjoill- 

kurlicher Funktionen einer variablen durch trigonometrische 
Beihen, Gottingen, 1879. 

80. Bois-Reymond, Paul du. Zur Oeschichte der Trigonometriscfien 

Beihen, Eine Entgegnung. TQbingen. 

81. PoiNGARE, Hexbi. Noticc sur les Travaux Scientifiques de Henri 

PoincarL Paris, 1886. 

82. Bjbrknbs, C. a. Niels-Henrik Abel, Tctbleau de sa vie et de son 

action scient^fique. Paris, 1885. 

83. TucKEB, R. ** Carl Friedrich Gauss," Nature, April, 1877. 

84. DiBicHLET, Lejeukb. Qcddchnissredc auf Carl Qustav Jacob 

Jacobi. 1862. 
86. Eithbpeb, Alfred. Elliptische Funktionen, Theorie und Ge- 
schichu. Halle a/S., 1876. 

86. Hbvbici, O. ** Theory of Functions," Nature, 43 : 14 and 16, 1891. 

87. Dabboux, Gaston. Notice sur les Travaux Scientiflques de M. Qas- 

ton Darboux. Paris, 1884. 

88. KuMMEB, E. £. Oeddchnissrede auf Qustav Peter Lejeune-Dirichlet, 

Berlin, 1860. 
80. Smith, H. J. Stephen. **0n the Present State and Prospects of 
Some Branches of Pure Mathematics," Proceedings of the London 
Mathematical Society, Vol. VIII., Nos. 104, 106, 1876. 

90. Glaisheb, J. W. L. * * Henry John Stephen Smith," Monthly Notices 

of the Boyal Astronomical Society, XLIV., 4, 1884. 

91. Bessel als Bremer Handlungslehrling. Bremen, 1890. 

92. Fbantz,J. Festredeaus VeranlassungvonBesseVshunder^dhrigem 

Geburtstag, Konigsberg, 1884. 

93. Dziobek, O. Mathematical Theories of Planetary Motions. 

Translated into English by M. W. Harrington and W. J. Hussey. 

94. HsBMrrE, Ch. ** Discours prononc6 devant le prteident de la R6pu- 

blique," Bulletin des Sciences Math^matiques, XIV., Janvier, 
1890. 

95. ScnusTEB, Abthub. **The Influence of Mathematics on the Prog- 

ress of Physics," Nature, 26 : 17, 1882. 

96. Kebredz, E. de. ** Sophie de Kowalevski," Bendiconti del Circolo 

Matematico di Palermo, V., 1891. 

97. Voiot, W. Zum Cfeddchniss von G. Kirchhoff, Gottingen, 1888. 



XIV BOOKS OF BEFEREKOE. 

08. BdcHBR, Maxime. ** a Bit of Mathematical History," Bulletin oj 

the N. r. Math. Soc, Vol. II., No. 5. 

09. Catley, Arthur. Beport on the Recent Progress of TheoretUal 

Dynamics, 1857. 

100. Glaze BROOK, R. T. Beport on Optical Theories. 1886. 

101. RosENBEROBR, F. Qcschichte der Physik. Braunschweig, 1887-1890. 



A HISTORY OF MATHEMATICS. 



INTRODUCTION. 

The contemplation of the various steps by which mankind 
has come into possession of the vast stock of mathematical 
knowledge can hardly fail to interest the mathematician. He 
takes pride in the fact that his science, more than any other, 
is an exact science, and that hardly anything ever done in 
mathematics has proved to be useless. The chemist smiles 
at the childish efforts of alchemists, but the mathematician 
finds the geometry of the Greeks and the arithmetic of the 
Hindoos as useful and admirable as any research of to-day. 
He is pleased to notice that though, in course of its develop- 
ment, mathematics has had periods of slow growth, yet in 
the main it has been pre-eminently a progressive science. 

The history of mathematics may be instructive as well as 
agreeable ; it may not only remind us of what we have, but 
may also teach us how to increase our store. Says De Morgan, 
" The early history of the mind of men with regard to mathe- 
matics leads us to point out our own errors; and in this 
respect it is well to pay attention to the history of mathe- 
matics." It warns us against hasty conclusions ; it points out 
the importance of a good notation upon the progress of the 
science ; it discourages excessive specialisation on the part of 

1 



2 A HISTOBY OF MATHEMATICS. 

investigators, by showing how apparently distinct branches 
have been found to possess unexpected connecting links; it 
saves the student from wasting time and energy upon prob- 
lems which were, perhaps, solved long since; it discourages 
him from attacking an unsolved problem by the same method 
which has led other mathematicians to failure ; it teaches that 
fortifications can be taken in other ways than by direct attack, 
that when repulsed from a direct assault it is well to recon- 
noitre and occupy the surrounding ground and to discover the 
secret paths by which the apparently unconquerable position 
can be taken.^ The importance of this strategic rule may 
be emphasised by citing a case in which it has been violated. 
An untold amount of intellectual energy has been expended 
on the quadrature of the circle, yet no conquest has been made 
by direct assault. The circle-squarers have existed in crowds 
ever since the period of Archimedes. After innumerable fail- 
ures to solve the problem at a time, even, when investigators 
possessed that most powerful tool, the differential calculus, 
persons versed in mathematics dropped the subject, while 
those who still persisted were completely ignorant of its his- 
tory and generally misunderstood the conditions of the prob- 
lem. "Our problem," says De Morgan, "is to square the 
circle with the old dUouxmce of means : Euclid's postulates 
and nothing more. We cannot remember an instance in which 
a question to be solved by a definite method was tried by the 
best heads, and answered at last, by that method, after thou- 
sands of complete failures." But progress was made on this 
problem by approaching it from a different direction and by 
newly discovered paths. Lambert proved in 1761 that the 
ratio of the circumference of a circle to its diameter is incom- 
mensurable. Some years ago, Lindemann demonstrated that 
this ratio is also transcendental and that the quadrature of 
the circle, by means of the ruler and compass only, is impos- 



INTRODUCTION. 8 

sibU. He thus showed by actual proof that which keen- 
minded mathematicians had long suspected ; namely, that the 
great army of circle-squarers have, for two thousand years, 
been assaulting a fortification which is as indestructible as 
the firmament of heaven. 

Another reason for the desirability of historical study is 
the value of historical knowledge to the teacher of mathe- 
matics. The interest which pupils take in their studies may 
be greatly increased if the solution of problems and the cold 
logic of geometrical demonstrations are interspersed with 
historical remarks and anecdotes. A cla^s in arithmetic will 
be pleased to hear about the Hindoos and their invention of 
the "Arabic notation"; they will marvel at the thousands 
of years which elapsed before people had even thought of 
introducing into the numeral notation that Columbus-egg — 
the zero; they will find it astounding that it should have 
taken so long to invent a notation which they themselves can 
now learn in a month. After the pupils have learned how to 
bisect a given angle, surprise them by telling of the many 
futile attempts which have been made to solve, by elementary 
geometry, the apparently very simple problem of the trisec- 
tion of an angle. When they know how to construct a square 
whose area is double the area of a given square, tell them 
about the duplication of the cube — how the wrath of Apollo 
could be appeased only by the construction of a cubical altar 
double the given altar, and how mathematicians long wrestled 
with this problem. After the class have exhausted their ener- 
gies on the theorem of the right triangle, tell them something 
about its discoverer — how Pythagoras, jubilant over his great 
accomplishment, sacrificed a hecatomb to the Muses who in- 
spired him. When the value of mathematical training is 
called in question, quote the inscription over the entrance into 
the academy of Plato, the philosopher : " Let no one who is 



4 A HISTORY OF MATHEBfATICS. 

unacquainted with geometry enter here." Students in analyt- 
ical geometry should know something of Descartes, and, after 
taking up the differential and integral calculus, they should 
become familiar with the parts that Newton, Leibniz, and 
Lagrange played in creating that science. In his historical 
talk it is possible for the teacher to make it plain to the 
student that mathematics is not a dead science, but a living 
one in which steady progress is made.' 

The history of mathematics is important also as a valuable 
contribution to the history of civilisation. Human progress 
is closely identified with scientific thought. Mathematical 
and physical researches are a reliable record of intellectual 
progress. The history of mathematics is one of the large 
windows through which the philosophic eye looks into past 
ages and traces the line of intellectual development. 



ANTIQUITY. 



THE BABYLONIANS. 

Thb fertile valley of the Euphrates and Tigris was one of 
the primeval seats of human society. Authentic history of 
the peoples inhabiting this region begins only with the foun- 
dation, in Chaldaea and Babylonia, of a united kingdom out 
of the previously disunited tribes. Much light has been 
thrown on their history by the discoveiy of the art of reading 
the cuneiform or wedge-shaped system of writing. 

In the study of Babylonian mathematics we begin with the 
notation of numbers. A vertical wedge V stood for 1, while 
the characters ^ and V^ signified 10 and 100 respec- 
tively. Grotefend believes the character for 10 originally to 
have been the picture of two hands, as held in prayer, the 
palms being pressed together, the fingers close to each other, 
but the thumbs thrust out. In the Babylonian notation two 
principles were employed — the additive and multiplicar 
tive. Numbers below 100 were expressed by symbols whose 
respective values had to be added. Thus, ^ Y stood for 2, 
J |[ y for 3, ▼^^ for 4, ^r^ for 23, ^ ^ ^ for 30. Here the 
symbols of higher order appear always to the left of those of 
lower order. In writing the hundreds, on the other hand, a 
tmaller symbol was placed to the left of the 100, and was, in 
that case, to be muUiplied by 100. Thus, ^ V ^ signified 



6 A HISTORY OP MATHEMATICS. 

10 times 100, or 1000. But this symbol for 1000 was itself 
taken for a new unit, which could take smaller coefficients to 
its left. Thus, ^ ^ ^ ^ denoted, not 20 times 100, but 
10 times 1000. Of the largest numbers written in cuneiform 
symbols, which have hitherto been found, none go as high as 
a million.' 

If, as is believed by most specialists, the early Sumerians 
were the inventors of the cuneiform writing, then they were, 
in all probability, also familiar with the notation of numbers. 
Most surprising, in this connection, is the fact that Sumerian 
inscriptions disclose the use, not only of the above decimal 
system, but also of a sexagesimal one. The latter was used 
chiefly in constructing tables for weights and measures. It is 
full of historical interest. Its consequential development, 
both for integers and fractions, reveals a high degree of 
mathematical insight. We possess two Babylonian tablets 
which exhibit its use. One of them, probably written between 
2300 and 1600 b.c, contains a table of square numbers up to 
60*. The numbers 1, 4, 9, 16, 25, 36, 49, are given as the 
squares of the first seven integers respectively. We have next 
1.4 = 8«, 1.21 = 9*, 1.40 = 10*, 2.1 = 11*, etc. This remains 
unintelligible, unless we assume the sexagesimal scale, which 
makes 1.4 = 60 + 4, 1.21 = 60 + 21, 2.1 = 2.60 -h 1. The second 
tablet records the magnitude of the illuminated portion of the 
moon's disc for every day from new to full moon, the whole disc 
being assumed to consist of 240 parts. The illuminated parts 
during the first five days are the series 5, 10, 20, 40, 1.20 
(=80), which is a geometrical progression. From here on 
the series becomes an arithmetical progression, the numbers 
from the fifth to the fifteenth day being respectively 1.20, 1.36, 
1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4. This table not 
only exhibits the use of the sexagesimal system, but also indi- 
cates the acquaintance of the Babylonians with progressions. 



THE BABYLONIANS. 7 

Not to be overlooked is the fact that in the sexagesimal nota- 
tion of integers the "principle of position" was employed. 
ThaSy in 1.4 (=64), the 1 is made to stand for 60, the unit 
of the second order, by virtue of its position with respect to 
the 4. The introduction of this principle at so early a date 
is the more remarkable, because in the decimal notation it 
was not introduced till about the fifth or sixth century after 
Christ. The principle of position, in its general and syste- 
matic application, requires a symbol for zero. We ask, Did 
the Babylonians possess one? Had they already taken the 
gigantic step of representing by a symbol the absence of 
units? Neither of the above tables answers this question, 
for they happen to contain no number in which there was 
occasion to use a zero. The sexagesimal system was used also 
in fractions. Thus, in the Babylonian inscriptions, ^ and ^ 
are designated by 30 and 20, the reader being expected, in 
his mind, to supply the word "sixtieths." The Greek geom- 
eter Hypsides and the Alexandrian astronomer Ptolemseus 
borrowed the sexagesimal notation of fractions from the 
Babylonians and introduced it into Greece. From that time 
sexagesimal fractions held almost full sway in astronomical 
and mathematical calculations until the sixteenth century, 
when they finally yielded their place to the decimal fractions. 
It may be asked. What led to the invention of the sexagesi- 
mal system? Why was it that 60 parts were selected? To 
this we have no positive answer. Ten was chosen, in the 
decimal system, because it represents the number of fingers. 
But nothing of the human body could have suggested 60. 
Cantor offers the following theory : At first the Babylonians 
reckoned the year at 360 days. This led to the division of 
the circle into 360 degrees, each degree representing the daily 
amount of the supposed yearly revolution of the sun around 
the earth. Now they were, very probably, familiar with the 



8 A HISTORY OF MATHEMATICS. 

fact that the radius can be applied to its circumference as a 
chord 6 times, and that each of these chords subtends an arc 
measuring exactly 60 degrees. Fixing their attention upon 
these degrees, the division into 60 parts may have suggested 
itself to them. Thus, when greater precision necessitated a 
subdivision of the degree, it was partitioned into 60 minutes. 
In this way the sexagesimal notation may have originated. 
The division of the day into 24 hours, and of the hour 
into minutes and seconds on the scale of 60, is due to the 
Babylonians. 

It appears that the people in the Tigro-Euphrates basin had 
made veiy creditable advance in arithmetic. Their knowledge 
of arithmetical and geometrical progressions has already been 
alluded to. lamblichus attributes to them also a knowledge 
of proportion, and even the invention of the so-called musical 
proportion. Though we possess no conclusive proof, we have 
nevertheless reason to believe that in practical calculation 
they used the dbacus. Among the races of middle Asia, even 
as far as China, the abacus is as old as fable. Now, Babylon 
was once a great commercial centre, — the metropolis of many 
nations, — and it is, therefore, not unreasonable to suppose that 
her merchants employed this most improved aid to calculation. 

In geometry the Babylonians accomplished almost nothing. 
Besides the division of the circumference into 6 parts by its 
radius, and into 360 degrees, they had some knowledge of 
geometrical figures, such as the triangle and quadrangle, which 
they used in their auguries. Like the Hebrews (1 Kin. 7 : 23), 
they took ir = 3. Of geometrical demonstrations there is, of 
course, no trace. ^' As a rule, in the Oriental mind the intui- 
tive powers eclipse the severely rational and logical." 

The astronomy of the Babylonians has attracted much 
attention. They worshipped the heavenly bodies from the 
earliest historic times. When Alexander the Great, after 



THE EGYPTIANS. 9 

the battle of Arbela (331 b.g.), took possession of Babylon, 
Callisthenes found there on burned brick astronomical records 
reaching back as far as 2234 b.c. Porphyrins says that these 
were sent to Aristotle. Ptolemy, the Alexandrian astrono- 
mer, possessed a Babylonian record of eclipses going back to 
747 B.C. Eecently Epping and Strassmaier ^ threw considera- 
ble light on Babylonian chronology and astronomy by explain- 
ing two calendars of the years 123 b.c. and 111 b.g., taken 
from cuneiform tablets coming, presumably, from an old 
obsenratory. These scholars have succeeded in giving an 
account of the Babylonian calculation of the new and full 
moon, and have identified by calculations the Babylonian 
names of the planets, and of the twelve zodiacal signs and 
twenty-eight normal stars which correspond to some extent 
with the twenty-eight naksJuUraa of the Hindoos. We append 
part of an Assyrian astronomical report, as translated by 
Oppert: — 

**To the King, my lord, thy faithfal servant, Mar-Istar.** 

** . . . On the first day, as the new moon^s day of the month Tham- 
muz declined, the moon was again visible over the planet Mercury, as 
I had already predicted to my master the King. I erred not.** 



THE EGYPTIANS. 

Though there is great difference of opinion regarding the 
antiquity of Egyptian civilisation, yet all authorities agree in 
the statement that, however far back they go, they find no 
uncivilised state of society. " Menes, the first king, changes 
the course of the Nile, makes a great reservoir, and builds the 
temple of Phthah at Memphis." The Egyptians built the 
at a very early period. Surely a people engaging in 



10 A HISTORY OF MATHEMATICS. 

enterprises of such magnitude must have known something of 
mathematics — at least of practical mathematics. 

All Greek writers are unanimous in ascribing, without 
envy, to Egypt the priority of invention in the mathematical 
sciences. Plato in Phcedrua says : '^ At the Egyptian city 
of Kaucratis there was a famous old god whose name was 
Theuth; the bird which is called the Ibis was sacred to 
him, and he was the inventor of many arts, such as arithmetic 
and calculation and geometry and astronomy and draughts 
and dice, but his great discovery was the use of letters." 

Aristotle says that mathematics had its birth in Egypt, 
because there the priestly class had the leisure needful for 
the study of it. Geometry, in particular, is said by Herodotus, 
Diodorus, Diogenes Laertius, lamblichus, and other ancient 
writers to have originated in Egypt.* In Herodotus we find 
this (II. c. 109) : " They said also that this king [Sesostris] 
divided the land among all Egyptians so as to give each one a 
quadrangle of equal size and to draw from each his revenues, 
by imposing a tax to be levied yearly. But every one from 
whose part the river tore away anything, had to go to him 
and notify what had happened; he then sent the overseers, 
who had to measure out by how much the land had become 
smaller, in order that the owner might pay on what was left, 
in proportion to the entire tax imposed. In this way, it 
appears to me, geometry originated, which passed thence to 
Hellas." 

We abstain from introducing additional Greek opinion 
regarding Egyptian mathematics, or from indulging in wild 
conjectures. We rest our account on documentary evidence. 
A hieratic papyrus, included in the Rhind collection of the 
British Museum, was deciphered by Eisenlohr in 1877, and 
found to be a mathematical manual containing problems in 
arithmetic and geometry. It was written by Ahmes some 



THE EGYPTIANS. 11 

time before 1700 b.c., and was founded on an older work 
believed by Birch to date back as far as 3400 b.c. ! This 
curious papyrus — the most ancient mathematical handbook 
known to us — puts us at once in contact with the mathe- 
matical thought in Egypt of three or five thousand years 
ago. It is entitled ** Directions for obtaining the Knowledge 
of all Dark Things." We see from it that the Egyptians 
cared but little for theoretical results. Theorems are not 
found in it at all. It contains " hardly any general rules of 
procedure, but chiefly mere statements of results intended 
possibly to be explained by a teacher to his pupils."^ In 
geometry the forte of the Egyptians lay in making con- 
structions and determining areas. The area of an isosceles 
triangle, of which the sides measure 10 ruths and the base 
4 ruths, was erroneously given as 20 square ruths, or half 
the product of the base by one side. The area of an isosceles 
trapezoid is found, similarly, by multiplying half the sum 
of the parallel sides by one of the non-parallel sides. The 
area of a circle is found by deducting from the diameter 
^ of its length and squaring the remainder. Here tr is 
taken = (J^)*= 3.1604 •••, a very fair approximation.* The 
papyrus explains also such problems as these, — To mark 
out in the field a right triangle whose sides are 10 and 4 
units; or a trapezoid whose parallel sides are 6 and 4, and 
the non-parallel sides each 20 units. 

Some problems in this papyrus seem to imply a rudi- 
mentary knowledge of proportion. 

The base-lines of the pyramids run north and south, and 
east and west, but probably only the lines running north and 
south were determined by astronomical observations. This, 
coupled with the fact that the word harpedonaptoe, applied to 
Egyptian geometers, means " rope-stretchers," would point to 
the conclusion that the Egyptian, like the Indian and Chinese 



12 A HISTOKY OF MATHEMATICS. 

geometers, constructed a right triangle upon a given line, by 
stretching around three pegs a rope consisting of three parts 
in the ratios 3:4:5, and thus forming a right triangle.' If 
this explanation is correct, then the Egyptians were familiar, 
2000 years B.C., with the well-known property of the right 
triangle, for the special case at least when the sides are 
in the ratio 3:4:5. 

On the walls of the celebrated temple of Horus at Edfu 
have been found hieroglyphics, written about 100 b.c, which 
enumerate the pieces of land owned by the priesthood, and 
give their areas. The area of any quadrilateral, however 

irregular, is there found by the formula "^ • "^ » Thus, 

for a quadrangle whose opposite sides are 5 and 8, 20 and 
15, is given the area 113^ \^ The incorrect formulae of 
Ahmes of 3000 years b.c. yield generally closer approxima- 
tions than those of the Edfu inscriptions, written 200 years 
after Euclid ! 

The fact that the geometry of the Egyptians consists 
chiefly of constructions, goes far to explain certain of its 
great defects. The Egyptians failed in two essential points 
without which a science of geometry, in the true sense of 
the word, cannot exist. In the first place, they failed to 
construct a rigorously logical system of geometry, resting 
upon a few axioms and postulates. A great many of their 
rules, especially those in solid geometry, had probably not 
been proved at all, but were known to be true merely from 
observation or as matters of fact. The second great defect 
was their inability to bring the numerous special cases under 
a more general view, and thereby to arrive at broader and 
more fundamental theorems. Some of the simplest geo- 
metrical truths were divided into numberless special cases 
of which each was supposed to require separate treatment. 



THE EGYPTIANS. 13 

Some paitictQars about Egyptian geometry can be men- 
tioned more advantageously in connection with the early 
Greek mathematicians who came to the Egyptian priests for 
iustmction. 

An insight into Egyptian methods of numeration was ob- 
tained through the ingenious deciphering of the hieroglyphics 
by Champollion, Young, and their successors. The symbols 
used were the following: I for 1, f\ for 10, (^ for 100, 
& for 1000, f for 10,000, ^ for 100,000, ^ for 1,000,000, 
,Q for 10,000,000.* The symbol for 1 represents a vertical 
staff; that for 10,000 a pointing finger; that for 100,000 a 
burbot; that for 1,000,000, a man in astonishment. The 
significance of the remaining symbols is very doubtful. 
The writing^f numbers with these hieroglyphics was very 
cumbrous. The unit symbol of each order was repeated as 
many times as there were units in that order. The principle 
employed was the additive. Thus, 23 was written fll O II I I 

Besides the hieroglyphics, Egypt possesses the hieratic and 
demotic writings, but for want of space we pass them by. 

Herodotus makes an important statement concerning the 
mode of computing among the Egyptians. He says that 
they '' caicuUUe with pebbles by moving the hand from right 
to left, while the Hellenes move it from left to right.'' 
Herein we recognise again that instrumental method of 
figuring so extensively used by peoples of antiquity. The 
Egyptians used the decimal scale. Since, in figuring, they 
moved their hands horizontally, it seems probable that they 
used eiphering-boards with vertical columns. In each column 
there must have been not more than nine pebbles, for ten 
pebbles would be equal to one pebble in the column next 
to the left 

The Ahmes papyrus contains interesting information on 
the way in which the Egyptians employed fractions. Their 



{ 



14 A HISTORY OF MATHEMATICS. 

methods of operation were, of course, radically different from 
ours. Fractions were a subject of very great difficulty with 
the ancients. Simultaneous changes in both numerator and 
denominator were usually avoided. In manipulating fractions 
the Babylonians kept the denominators (60) constant. The 
Romans likewise kept them constant, but equal to 12. The 
Egyptians and Greeks, on the other hand, kept the numerators 
constant, and dealt with variable denominators. Ahmes used 
the term ^' fraction " in a restricted sense, for he applied it 
only to unit-fractionSf or fractions having unity for the numer- 
ator. It was designated by writing the denominator and then 
placing over it a dot. Fractional values which could not be 
expressed by any one unit-fraction were expressed as the 
sum of two or more of them. Thus, he wrote ^ -jiy in place 
of J. The first important problem naturally arising was, how 
to represent any fractional value as the sum of unit-fractions. 
This was solved by aid of a table, given in the papyrus, in 

2 

which all fractions of the form (where n designates 

successively all the numbers up to 49) are reduced to the 
sum of unit-fractions. Thus, ^ = J ^ ; A = A" rir- When, 
by whom, and how this table was calculated, we do not know. 
Probably it was compiled empirically at different times, by 
different persons. It will be seen that by repeated applica- 
tion of this table, a fraction whose numerator exceeds two 
can be expressed in the desired form, provided that there 
is a fraction in the table having the same denominator that 
it has. Take, for example, the problem, to divide 6 by 21. 
In the first place, 6 = 1 -f 2 -f 2. From the table we get 

^^^ij. Then3^ = ^-f(TJ^:^) + (VtA) = ^-h(AA) 
= ^^^ = ^^ = ^-j^^. The papyrus contains problems 

in which it is required that fractions be raised by addition 

or multiplication to given whole numbers or to other fractions. 



THE EGYPTIANS. 15 

For example, it is required to increase J i iV tAt A *^ 1- 
The common denominator taken appears to be 45, for the 
numbers are stated as 11^^, 6J \, 4J-, 1^ 1. The sum of these 
is 23J \ \ forty-fifths. Add to this ^ -^^ and the sum is |. 
Add ^, and we have 1. Hence the quantity to be added to 
the given fraction is ^ ^ ^. 

Having finished the subject of fractions, Ahmes proceeds to 
the solution of equations of one unknown quantity. The un- 
known quantity is called 'hau' or heap. Thus the problem, 

"heap, its |, its whole, it makes 19," i.e. - -f a? = 19. In this 

case, the solution is as follows : -;^ = 19 ; ? = 2 J J^ ; a? = 16^ \. 

But in other problems, the solutions are effected by various 
other methods. It thus appears that the beginnings of algebra 
are as ancient as those of geometry. 

The principal defect of Egyptian arithmetic was the lack of 
a simple, comprehensive symbolism — a defect which not even 
the Greeks were able to remove. 

The Ahmes papyrus doubtless represents the most advanced 
attainments of the Egyptians in arithmetic and geometry. It 
is remarkable that they should have reached so great profi- 
ciency in mathematics at so remote a period of antiquity. But 
strange, indeed, is the fact that, during the next two thousand 
years, they should have made no progress whatsoever in it. 
The conclusion forces itself upon us, that they resemble the 
Chinese in the stationary characteVy not only of their govern- 
ment, but also of their learning. All the knowledge of geom- 
etry which they possessed when Greek scholars visited them, 
six centuries b.c., was doubtless known to them two thousand 
years earlier, when they built those stupendous and gigantic 
structures — the pyramids. An explanation for this stagnation 
of learning has been sought in the fact that their early dis- 
ooreries in mathematics and medicine had the misfortune of 



i 

16 A HISTORY OF MATHEMATICS. 

being entered upon their sacred books and that, in after ages, 
it was considered heretical to augment or modify anything 
therein. Thus the books themselves closed the gates to 
progress. 

THE GREEKS. 

About the seventh century b.c. an active commercial inter- 
course sprang up between Greece and Egypt. Naturally there 
arose an interchange of ideas as well as of merchandise. 
Greeks, thirsting for knowledge, sought the Egyptian priests 
for instruction. Thales, Pythagoras, (Enopides, Plato, De- 
mocritus, Eudoxus, all visited the land of the pyramids. 
Egyptian ideas were thus transplanted across the sea and 
there stimulated Greek thought, directed it into new lines, 
and gave to it a basis to work upon. Greek culture, therefore, 
is not primitive. Not only in mathematics, but also in my- 
thology and art, Hellas owes a debt to older coimtries. To 
Egypt Greece is indebted, among other things, for its element- 
ary geometry. But this does not lessen our admiration for 
the Greek mind. From the moment that Hellenic philoso- 
phers applied themselves to the study of Egyptian geometry, 
this science assumed a radically different aspect. '< Whatever 
we Greeks receive, we improve and perfect," says Plato. The 
Egyptians carried geometry no further than was absolutely 
necessary for their practical wants. The Greeks, on the other 
hand, had within them a strong speculative tendency. They 
felt a craving to discover the reasons for things. They found 
pleasure in the contemplation of ideal relations, and loved 
science da science. 

Our sources of information on the history of Greek geometry 
before Euclid consist merely of scattered notices in ancient 
writers. The early mathematicians, Thales and Pythagoras, 



THE GREEKS. 17 

left behind no written records of their discoveries. A full 
history of Greek geometry and astronomy during this period, 
written by Eudemus, a pupil of Aristotle, has been lost. It 
was well known to Proclus, who, in his commentaries on 
Eoclid, gives a brief account of it. This abstract constitutes 
our most reliable information. We shall quote it frequently 
under the name of Eudemian Summary. 

The Ionic School. 

To Thalea of Miletus (640-546 b.c.), one of the ''seven wise 
men,'' and the founder of the Ionic school, falls the honour of 
having introduced the study of geometry into Greece. During 
middle life he engaged in commercial pursuits, which took 
him to Egypt. He is said to have resided there, and to have 
studied the physical sciences and mathematics with the Egyp- 
tian priests. Plutarch declares that Thales soon excelled his 
masters, and amazed King Amasis by measuring the heights 
of the pyramids from their shadows. According to Plutarch, 
this was done by considering that the shadow cast by a verti- 
cal staff of known length bears the same ratio to the shadow 
of the pyramid as the height of the staff bears to the height 
of the pyramid. This solution presupposes a knowledge of 
proportion, and the Ahmes papyrus actually shows that the 
rudiments of proportion were known to the Egyptians. Ac- 
cording to Diogenes Laertius, the pyramids were measured by 
Thales in a different way ; viz. by finding the length of the 
shadow of the pyramid at the moment when the shadow of a 
staff was equal to its own length. 

The Eudemian Summary ascribes to Thales the invention 
of the theorems on the equality of vertical angles, the equality 
of the angles at the base of an isosceles triangle, the bisec- 
tion of a circle by any diameter, and the congruence of two 



18 A HISTOBT OF MATHEMATICS. 

triangles having a side and the two adjacent angles equal re- 
spectively. The last theorem he applied to the measurement 
of the distances of ships from the shore. Thus Thales was 
the first to apply theoretical geometry to practical uses. The 
theorem that all angles inscribed in a semicircle are right 
angles is attributed by some ancient writers to Thales, by 
others to Pythagoras. Thales was doubtless familiar with 
other theorems, not recorded by the ancients. It has been 
inferred that he knew the sum of the three angles of a tri- 
angle to be equal to two right angles, and the sides of equi- 
angular triangles to be proportional.^ The Egyptians must 
have made use of the above theorems on the straight line, in 
some of their constructions found in the Ahmes papyrus, but 
it was left for the Greek philosopher to give these truths, 
which others saw, but did not formulate into words, an 
explicit, abstract expression, and to put into scientific lan- 
guage and subject to proof that which others merely felt to 
be true. Thales may be said to have created the geometry 
of lines, essentially abstract in its character, while the Egyp- 
tians studied only the geometry of surfaces and the rudiments 
of solid geometry, empirical in their character.' 

With Thales begins also the study of scientific astronomy. 
He acquired great celebrity by the prediction of a solar eclipse 
in 586 B.C. Whether he predicted the day of the occurrence, 
or simply the year, is not known. It is told of him that 
while contemplating the stars during an evening walk, he fell 
into a ditch. The good old woman attending him exclaimed, 
" How canst thou know what is doing in the heavens, when 
thou seest not what is at thy feet ?'' 

The two most prominent pupils of Thales were Anazimander 
(b. 611 B.C.) and Anaximenes (b. 570 b.c.). They studied 
chiefly astronomy and physical philosophy. Of Anazagoras, a 
pupil of Anaximenes, and the last philosopher of the Ionic 



THE GREEKS. 19 

school, we know little, except that, while in prison, he passed 
his time attempting to square the circle. This is the first 
time, in the history of mathematics, that we find mention of 
the famous problem of the quadrature of the circle, that rock 
upon which so many reputations have been destroyed. It 
turns upon the determination of the exact value of v. Approx- 
imations to w had been made by the Chinese, Babylonians, 
Hebrews, and Egyptians. But the invention of a method to 
find its exact value, is the knotty problem which has engaged 
the attention of many minds from the time of Anaxagoras 
down to our own. Anaxagoras did not offer any solution of 
it, and seems to have luckily escaped paralogisms. 

About the time of Anaxagoras, but isolated from the Ionic 
school, flourished (Enopides of Chios. Proclus ascribes to him 
the solution of the following problems : From a point without, 
to draw a perpendicular to a given line, and to draw an angle 
<m a line equal to a given angle. That a man could gain a 
reputation by solving problems so elementary as these, indi- 
cates that geometry was still in its infancy, and that the 
Greeks had not yet gotten far beyond the Egyptian con- 
stractions. 

The Ionic school lasted over one hundred years. The 
progress of mathematics during that period was slow, as 
compared with its growth in a later epoch of Greek history. 
A new impetus to its progress was given by Pythagoras. 

The School of Pythagoras, 

Pythagoras (580?^500? b.c.) was one of those figures which 
impressed the imagination of succeeding times to such an 
extent that their real histories have become difficult to be 
discerned through the mythical haze that envelops them. The 
following account of Pythagoras excludes the most doubtful 



20 A HISTORY OF MATHEMATICS. 

statements. He was a native of Samos, and was drawn by 
the fame of Pherecydes to the island of Syros. He then 
visited the ancient Thales, who incited him to study in Egypt. 
He sojourned in Egypt many years, and may have visited 
Babylon. On his return to Samos, he found it under the 
tyranny of Polycrates. Failing in an attempt to found a 
school there, he quitted home again and, following the current 
of civilisation, removed to Magna Orsecia in South Italy. He 
settled at Croton, and founded the famous Pythagorean school. 
This was not merely an academy for the teaching of philosophy, 
mathematics, and natural science, but it was a brotherhood, 
the members of which were united for life. This brotherhood 
had observances approaching masonic peculiarity. They were 
forbidden to divulge the discoveries and doctrines of their 
school. Hence we are obliged to speak of the Pythagoreans 
as a body, and find it difficult to determine to whom each 
particular discovery is to be ascribed. The Pythagoreans 
themselves were in the habit of referring every discovery back 
to the great founder of the sect. 

This school grew rapidly and gained considerable political 
ascendency. But the mystic and secret observances, intro- 
duced in imitation of Egyptian usages, and the aristocratic 
tendencies of the school, caused it to become an object of 
suspicion. The democratic party in Lower Italy revolted and 
desttored the buildings of the Pythagorean school Pythago- 
ras fled to Tarentum and thence to Metapontum, where he was 
murdered. 

Pythagoras has left behind no mathematical treatises, and 
our sources of information are rather scanty. Certain it is 
that, in the Pythagorean school, mathematics was the principal 
study. Pythagoras raised mathematics to the rank of a science. 
Arithmetic was courted by him as fervently as geometry. In 
fact, arithmetic is the foundation of his philosophic system. 



THE GREEKS. 21 

The Eudemian Summary says that '^ Pythagoras changed 
the study of geometiy into the form of a liberal education, 
for he examined its principles to the bottom, and investigated 
its theorems in an immaterial and intellectual manner.'' His 
geometry was connected closely with his arithmetic. He was 
especially fond of those geometrical relations which admitted 
of arithmetical expression. 

Like Egyptian geometry, the geometry of the Pythagoreans 
is much concerned with areas. To Pythagoras is ascribed the 
important theorem that the square on the hypotenuse of a 
right triangle is equal to the sum of the squares on the other 
two sides. He had probably learned from the Egyptians the 
troth of the theorem in the special case when the sides are 
3, 4, 5, respectively. The story goes, that Pythagoras was so 
jubilant over this discovery that he sacrificed a hecatomb. Its 
aathenticity is doubted, because the Pythagoreans believed in 
the transmigration of the soul and opposed, therefore, the 
shedding of blood. In the later traditions of the Neo-Pythago- 
reans this objection is removed by replacing this bloody sacri- 
ficeiby that of '^ an ox made of flour " ! The proof of the law 
of three squares, given in Euclid's Elements^ I. 47, is due to 
Euclid himself, and not to the Pythagoreans. What the Py- 
thagorean method of proof was has been a favourite topic for 
conjecture. 

The theorem on the sum of the three angles of a triangle, 
presumably known to Thales, was proved by the Pythagoreans 
after the manner of Euclid. They demonstrated also that the 
plane about a point is completely filled by six equilateral 
triangles, four squares, or three regular hexagons, so that it 
is possible to divide up a plane into figures of either kind. 

From the equilateral triangle and the square arise the solids, 
namely the tetraedron, octaedron, icosaedron, and the cube. 
These solids were, in all probability, known to the Egyptians, 



22 A HISTORY OF MATHEMATICS. 

excepting, perhaps, the icosaedron. In Pythagorean philos- 
ophy, they represent respectively the four elements of the 
physical world; namely, fire, air, water, and earth. Later 
another regular solid was discovered, namely the dodecaedron, 
which, in absence of a fifth element, was made to represent 
the universe itself. lamblichus states that Hippasus, a Py- 
thagorean, perished in the sea, because he boasted that he first 
divulged " the sphere with the twelve pentagons." The star- 
shaped pentagram was used as a symbol of recognition by the 
Pythagoreans, and was called by them Health. 

Pythagoras called the sphere the most beautiful of all solids, 
and the circle the most beautiful of all plane figures. The 
treatment of the subjects of proportion and of irrational 
quantities by him and his school will be taken up under the 
head of arithmetic. 

According to Eudemus, the Pythagoreans invented the prob- 
lems concerning the application of areas, including the cases 
of defect and excess, as in Euclid, VI. 28, 29. 

They were also familiar with the construction of a polygon 
equal in area to a given polygon and similar to another given 
polygon. This problem depends upon several important and 
somewhat advanced theorems, and testifies to the fact that 
the Pythagoreans made no mean progress in geometry. 

Of the theorems generally ascribed to the Italian school, 
some cannot be attributed to Pythagoras himself, nor to his 
earliest successors. The progress from empirical to reasoned 
solutions must, of necessity, have been slow. It is worth 
noticing that on the circle no theorem of any importance was 
discovered by this school. 

Though politics broke up the Pythagorean fraternity, yet 
the school continued to exist at least two centuries longer. 
Among the later Pythagoreans, Philolaus and Archytas are 
the most prominent. Philolaus wrote a book on the Pythago- 



THE GREEKS. 28 

rean doctrines. By him were first given to the world the 
teachings of the Italian school, which had been kept secret 
for a whole century. The brilliant Archytas of Tarentum 
(428-347 B.C.), known as a great statesman and general, and 
aniyersally admired for his virtues, was the only great geome- 
ter among the Greeks when Plato opened his school. Archy- 
tas was the first to apply geometry to mechanics and to treat 
the latter subject methodically. He also found a very ingeni- 
ous mechanical solution to the problem of the duplication of 
the cube. His solution involves clear notions on the genersr 
tion of cones and cylinders. This problem reduces itself to 
finding two mean proportionals between two given lines. 
These mean proportionals were obtained by Archytas from 
the section of a half-cylinder. The doctrine of proportion 
was advanced through him. 

There is every reason to believe that the later Pythagoreans 
exercised a strong influence on the study and development of 
mathematics at Athens. The Sophists acquired geometry from 
Pythagorean sources. Plato bought the works of Philolaus, 
and had a warm friend in Archytas. 

The Sophist School. 

After the defeat of the Persians under Xerxes at the battle 
of Salamis, 480 b.c, a league was formed among the Greeks 
to preserve the freedom of the now liberated Greek cities on 
the islands and coast of the ^gsean Sea. Of this league 
Athens soon became leader and dictator. She caused the 
separate treasury of the league to be merged into that of 
Athens, and then spent the money of her allies for her own 
aggrandisement. Athens was also a great commercial centre. 
Thus she became the richest and most beautiful city of an- 
tiquity. All menial work was performed by slaves. The 



24 A HISTORY OF MATHEMATICS. 

citizen of Athens was well-to-do and enjoyed a large amount 
of leisure. The government being purely democratic, every 
citizen was a politician. To make his influence felt among 
his fellow-men he must, first of all, be educated. Thius there 
arose a demand for teachers. The supply came principally 
from Sicily, where Pythagorean doctrines had spread. These 
teachers were called Sophists, or "wise men." Unlike the 
Pythagoreans, they accepted pay for their teaching. Although 
rhetoric was the principal feature of their instruction, they 
also taught geometry, astronomy, and philosophy. Athens 
soon became the headquarters o^ Grecian men of letters, and 
of mathematicians in particular. The home of mathematics 
among the Greeks was first in the Ionian Islands, then in 
Lower Italy, and during the time now under consideration, 
at Athens. 

The geometry of the circle, which had been entirely 
neglected by the Pythagoreans, was taken up by the Sophists. 
Nearly all their discoveries were made in connection with 
their innumerable attempts to solve the following three 
famous problems : — 

(1) To trisect an arc or an angle. 

(2) To " double the cube," i.e. to find a cube whose volume 
is double that of a given cube. 

(3) To "square the circle," i.e, to find a square or some 
other rectilinear figure exactly equal in area to a given circle. 

These problems have probably been the subject of more 
disciussion and research than any other problems in mathe- 
matics. The bisection of an angle was one of the easiest 
problems in geometry. The trisection of an angle, on the 
other hand, presented unexpected difficulties. A right angle 
had been divided into three equal parts by the Pjrthagoreans. 
But the general problem, though easy in appearance, tran- 
scended the power of elementary geometry. Among the first 



THE 6BBEEB. 25 

to wrestle with it was Hippias of EUb, a contemporary of 
Socrates, and bom about 460 b.c. Like all the later geome> 
ters, he &iled in effecting the trisection by means of a ruler 
and compass only. Proclus mentions a man, Hippias, presum- 
ably Hippias of Elis, as the inventor of a transcendental curve 
which served to divide an angle not only into three, but into 
any number of equal parts. This same curve was lused later 
by Deinostratus and others for the quadrature of the circle. 
On this account it is called the quadratrix. 

The Pythagoreans had shown that the diagonal of a square 
is the side of another square having double the area of the 
ohgiDal one. This probably suggested the problem of the 
duplication of the cube, i.e. to find the edge of a cube having 
double the volume of a given cube. Eratosthenes ascribes to 
this problem a different origin. The Delians were once suf- 
fering from a pestilence and were ordered by the oracle to 
double a certain cubical altar. Thoughtless workmen, simply 
constructed a cube with edges twice as long, but this did not 
pacify the gods. The error being discovered, Plato was con- 
sulted on the matter. He and his disciples searched eagerly 
for a solution to this '^ Delian Problem." Hippocrates of Chios 
(about 430 b.c.), a talented mathematician, but otherwise slow 
and stupid, was the first to show that the problem could be 
reduced to finding two mean proportionals between a given 
line and another twice as long. For, ;n the proportion a : x 
ss X : y = y : 2 a, since t? = ay and ^ = 2ax and «* = ay, we 
have a5* = 2 a'a? and oj* = 2 a*. But he failed to find the two 
mean proportionals. His attempt to square the circle was 
also a failure; for though he made himself celebrated by 
squaring a lune, he committed an error in attempting to apply 
this result to the squaring of the circle. 

In his study of the quadrature and duplication-problems, 
Hippocrates contributed much to the geometry of the circle. 




26 A HISTORY OF MATHEMATICS. 

The subject of similar figures was studied and partly- 
developed by Hippocrates. This involved the theory of 
proportion. Proportion had, thus far, been used by the 
Greeks only in numbers. They never succeeded in uniting 
the notions of numbers and magnitudes. The term "number *' 
was used by them in a restricted sense. What we call 
irrational numbers was not included under this notion. Not 
even rational fractious were called numbers. They used the 
word in the same sense as we use "integers." Hence num- 
bers were conceived as discontinuouSy while magnitudes were 
continuous. The two notions appeared, therefore, entirely 
distinct. The chasm between them is exposed to full view 
in the statement of Euclid that "incommensurable magni- 
tudes do not have the same ratio as numbers.'' In Euclid's 
Elements we find the theory of proportion of magnitudes 
developed and treated independent of that of numbers. The 
transfer of the theory of proportion from numbers to mag- 
nitudes (and to lengths in particular) was a difficult and 
important step. 

Hippocrates added to his fame by writing a geometrical 
text-book, called the Elements, This publication shows that 
the Pythagorean habit of secrecy was being abandoned; 
secrecy was contrary to the spirit of Athenian life. 

The Sophist Antiphon, a contemporary of Hippocrates, intro- 
duced the process of exhaustion for the purpose of solving 
the problem of the quadrature. He did himself credit by 
remarking that by inscribing in a circle a square, and on its 
sides erecting isosceles triangles with their vertices in the 
circumference, and on the sides of these triangles erecting 
new triangles, etc., one could obtain a succession of regular 
polygons of 8, 16, 32, 64 sides, and so on, of which each 
approaches nearer to the circle than the previous one, until 
the circle is finally exhausted. Thus is obtained an inscribed 



THE GREEKS. 27 

poljgon whose sides coincide with the circumference. Since 
there can be found squares equal in area to any polygou, 
there also can be found a square equal to the last polygon 
inscribed, and therefore equal to the circle itself. Bryson 
if Heradea, a contemporary of Antiphon, advanced the prob- 
lem of the quadrature considerably by circumscribing poly- 
gons at the same time that he inscribed polygons. He erred, 
boweyer, in assuming that the area of a circle was the arith- 
netical mean between circumscribed and inscribed polygons. 
Unlike Bryson and the rest of Greek geometers, Antiphou 
leems to have believed it possible, by continually doubling 
the sides of an inscribed polygon, to obtain a polygon coin- 
ciding with the circle. This question gave rise to lively 
disputes in Athens. If a polygon can coincide with the 
cirele, then, says Simplicius, we must put aside the notion 
that magnitudes are divisible ad infinitum. Aristotle always 
w ppqrtg i the theory of the infinite divisibility, while Zeno, ^^^ 

f ^tfTStoic^ ttempted to show its absurdity by proving that \La< ULi 
if magnitudes are infinitely divisible, motion is impossible. 
Zeno argues that Achilles could not overtake a tortoise ; for 
while he hastened to the place where the tortoise had been 
when he started, the tortoise crept some distance ahead, and 
while Achilles reached that second spot, the tortoise again 
nu>red forward a little, and so on. Thus the tortoise was 
^ways in advance of Achilles. Such arguments greatly con- 
founded Greek geometers. No wonder they were deterred 
^ such paradoxes from introducing the idea of infinity into 
^ir geometry. It did not suit the rigour of their proofs. 

The process of Antiphon and Bryson gave rise to the cum- 
"'"'^ns but perfectly rigorous "method of exhaustion." In 
^^(ennining the ratio of the areas between two curvilinear 
plane figures, say two circles, geometers first inscribed or 
^^^''camscribed similar polygons, and then by increasing indefi- 



28 A HISTORY OF MATHBKATIOS. 

nitely the number of sides, nearly exhausted the spaces 
between the polygons and circumferences. From the theo- 
rem that similar polygons inscribed in circles are to each 
other as the squares on their diameters, geometers may have 
divined the theorem attributed to Hippocrates of Chios that 
the circles, which differ but little from the last drawn poly- 
gons, must be to each other as the squares on their diameters. 
But in order to exclude all vagueness and possibility of doubt, 
later Greek geometers applied reasoning like that in Euclid, 
XII. 2, as follows : Let G and c, D and d be respectively the 
circles and diameters in question. Then if the proportion 
Z)» : cP = C7 : c is not true, suppose that Z)* : cP = O : c'. If c' < c, 
then a polygon p can be inscribed in the circle c which comes 
nearer to it in area than does c^ If P be the corresponding 
polygon in O, then P : p = Z>* : cP = O : c', and P:C = p : c*. 
Since p > c', we have P>Cy which is absurd. Next they 
proved by this same method of reductio ad abaurdum the 
falsity of the supposition that c' > c Since & can be neither 
larger nor smaller than c, it must be equal to it, q.e.d. 
Hankel refers this Method of Exhaustion back to Hippo- 
crates of Chios, but the reasons for assigning it to this early 
writer, rather than to Eadoxus, seem insufficient. 

Though progress in geometry at this period is traceable only 
at Athens, yet Ionia, Sicily, Abdera in Thrace, and Gyrene 
produced mathematicians who made creditable contributions 
to the science. We can mention here only Democritns of 
Abdera (about 460-370 b.c), a pupil of Anaxagoras, a friend 
of Philolaius, and an admirer of the Pjrthagoreans. He 
visited Egypt and perhaps even Persia. He was a successful 
geometer and wrote on incommensurable lines, on geometry, 
on numbers, and on perspective. None of these works are 
extant. He used to boast that in the construction of plane 
figures with proof no one had yet surpassed him, not even 



THB GREEKS. 29 

the so-called harpedonapto ('< rope-stretchers'') of Egypt. By 
this assertion he pays a flattering compliment to the skill 
and ability of the Egyptians. 

The Platonic School 

Daring the Peloponnesian War (431-404 b.c.) the progress 
of geometry was checked. After the war, Athens sank into 
the background as a minor political power, but advanced more 
and moie to the front as the leader in philosophy, literature, 
ftnd science. Plato was bom at Athens in 429 b.c, the year 
of the great plague, and died in 348. He was a pupil and 
near friend of Socrates, but it was not from him that he 
acquired his taste for mathematics. After the death of Soc- 
rates, Plato travelled extensively. In Cyrene he studied 
miihematics under Theodorus. He went to Egypt, then to 
I^>wer Italy and Sicily, where he came in contact with the 
Pythagoreans. Archytas of Tarentum and Timseius of Locri 
beeame his intimate friends. On his return to Athens, about 
3S9 B.C., he founded his school in the groves of the Academia, 
and devoted the remainder of his life to teaching and writing. 
Plato's physical philosophy is partly based on that of the 
Pythagoreans. Like them, he sought in arithmetic and 
geometry the key to the universe. When questioned about 
th« oocapation of the Deity, Plato answered that " He geom- 
ctrises continually." Accordingly, a knowledge of geometry 
is a necessary preparation for the study of philosophy. To 
^^ how great a value he put on mathematics and how 
neoeasaiy it is for higher speculation, Plato placed the inscrip- 
tioD over his porch, "Let no one who is unacquainted with 
geometry enter here." Xenocrates, a successor of Plato as 
teacher in the Academy, followed in his master's footsteps, by 
declining to admit a pupil who had no mathematical training. 



80 A HISTORY OF MATHBKATIGS. 

with the remark, ^< Depart, for thou hast not the grip of 
philosophy." Plato observed that geometry trained the mind 
for correct and vigorous thinking. Hence it was that the 
Eudemian Summary says, ^' He filled his writings with mathe- 
matical discoveries, and exhibited on every occasion the re- 
markable connection between mathematics and philosophy." 

With Plato as the head-master, we need not wonder that 
the Platonic school produced so large a number of mathemati- 
cians. Plato did little real original work, but he made 
valuable improvements in the logic and methods employed 
in geometry. It is true that the Sophist geometers of the 
previous century were rigorous in their proofs, but as a rule 
they did not reflect on the inward nature of their methods. 
They used the axioms without giving them explicit expression, 
and the geometrical concepts, such as the point, line, surface, 
etc., wMihout assigning to them formal definitions. The Py- 
thagoreans called a point '< unity in position," but this is a 
statement of a philosophical theory rather than a definition. 
Plato objected to calling a point a " geometrical fiction." He 
defined a point as the '^ beginning of a line " or as ''an indivis- 
ible line," and a line as " length without breadth." He called 
the point, line, surface, the ' boundaries ' of the line, surface, 
solid, respectively. Many of the definitions in Euclid are to 
be ascribed to the Platonic school. The same is probably 
true of Euclid's axioms. Aristotle refers to Plato the axiom 
that " equals subtracted from equals leave equals." 

One of the greatest achievements of Plato and his school is 
the invention of analysis as a method of proof. To be sure, 
this method had been used unconsciously by Hippocrates and 
others ; but Plato, like a true philosopher, turned the instinc- 
tive logic into a conscious, legitimate method. 

The terms syvthesis and analysis are used in mathematics 
in a more special sense than in logic. In ancient mathematics 



THE GREEKS. 31 

they had a different meaning from what they now have. The 
oldest definition of mathematical analysis as opposed to syn- 
thesis is that given in Euclid, XIII. 5, which in all probability 
wis framed by Eudoxus : " Analysis is the obtaining of the 
thing sought by assuming it and so reasoning up to an 
admitted truth; synthesis is the obtaining of the thing 
•oa^t by reasoning up to the inference and proof of it/' 
The analytic method is not conclusive, unless all operations 
involved in it are known to be reversible. To remove all 
doubt, the Greeks, as a rule, added to the analytic process 
a synthetic one, consisting of a reversion of all operations 
ooenrring in the analysis. Thius the aim of analysis was to 
aid in the discovery of synthetic proofs or solutions. 

Plato is said to have solved the problem of the duplication 
of the cube. But the solution is open to the very same objec- 
tioD which he made to the solutions by Archytas, Eudoxus, 
and KensBchmus. He called their solutions not geometrical, 
hot mechanical, for they required the use of other instruments 
than the ruler and compass. He said that thereby ^< the good 
of geometry is set aside and destroyed, for we again reduce it 
to the world of sense, instead of elevating and imbuing it with 
^ eternal and incorporeal images of thought, even as it is 
employed by God, for which reason He always is God." These 
objections indicate either that the solution is wrongly attrib- 
^ to Plato or that he wished to show how easily non-geo- 
metric solutions of that character can be found. It is now 
Seoerally admitted that the duplication problem, as well as 
the trisection and quadrature problems, cannot be solved by 
means of the ruler and compass only. 

Plato gave a healthful stimiilus to the study of stereometry, 
which until his time had been entirely neglected. The sphere 
^ the regular solids had been studied to some extent, but 
te prism, pyramid, cylinder, and cone were hardly known to 



82 A HISTORY OF MATHEMATICS. 

exist. All these solids became the subjects of inyestigation 
by the Platonic school. One result of these inquiries was 
epoch-making. Menschmus, an associate of Plato and pupil 
of Eudoxus, invented the conic sections, which, in course of 
only a century, raised geometry to the loftiest height which 
it was destined to reach during antiquity. Mensechmus cut 
three kinds of cones, the < right-angled,' 'acute-angled,' and 
'obtuse-angled,' by planes at right angles to a side of the 
cones, and thus obtained the three sections which we now call 
the parabola, ellipse, and hyperbola. Judging from the two 
very elegant solutions of the " Delian Problem " by means of 
intersections of these curves, Mensechmus must have succeeded 
well in investigating their properties. 

Another great geometer was Dinostratus, the brother of 
Mensechmus and pupil of Plato. Celebrated is his mechanical 
solution of the quadrature of the circle, by means of the quad- 
ratrix of Hippias. 

Perhaps the most brilliant mathematician of this period was 
EudoxuB. He was born at Gnidus about 408 b.c, studied under 
Archytas, and later, for two months, under Plato. He was 
imbued with a true spirit of scientific inquiry, and has been 
called the father of scientific astronomical observation. From 
the fragmentary notices of his astronomical researches, found 
in later writers, Ideler and Schiaparelli succeeded in recon- 
structing the system of Eudoxus with its celebrated representa- 
tion of planetary motions by " concentric spheres." Eudoxus 
had a school at Cyzicus, went with his pupils to Athens, visit- 
ing Plato, and then returned to Cyzicus, where he died 355 
B.C. The fame of the academy of Plato is to a large extent 
due to Eudoxus's pupils of the school at Cyzicus, among 
whom are Mensechmius, Dinostratus, Athenseus, and Helicon. 
Diogenes Laertius describes Eudoxus as astronomer, physician, 
legislator, as well as geometer. The Eudemian Summary 



THB GBBEKS. SS 

says that Eudozus ''first increased the number of general 
theorems, added to the three proportions three more, and 
raised to a considerable quantity the learning, begun by Plato, 
on the subject of the section, to which he applied the analyt- 
ical method." By this 'section' is meant, no doubt, the 
''golden section" (aectio aurea), which cuts a line in extreme 
aod mean ratio. The first five propositions in Euclid XIII. 
relate to lines cut by this section, and are generally attributed 
to Eudozus. Eudoxu9 added much to the knowledge of solid 
geometiy. He proved, says Archimedes, that a pyramid is 
exactly one-third of a prism, and a cone one-third of a cylinder, 
baying equal base and altitude. The proof that spheres are i 
to each other as the cubes of their radii is probably due to 
him. He made frequent and skilful use of the method of 
exhaustion, of which he was in all probability the inventor. 
A scholiast on Euclid, thought to be Froclus, says further that 
Eudozus practically invented the whole of Euclid's fifth book. . 
Eudozus also found two mean proportionals between two I 
given lines, but the method of solution is not known. ^ 

Plato has been called a maker of mathematicians. Besides 

the pupils already named, the Eudemian Summary men- 

tbns the followii^: Theaetetus of Athens, a man of great 

natural gifts, to whom, no doubt, Euclid was greatly indebted 

in the composition of the 10th book,^ treating of incommensu- 

nd>le8; Leodamas of Thasos; Neocleidea and his pupil Leon, 

who added much to the work of their predecessors, for Leon 

vrote an Elements carefully designed, both in number and 

utility of its proofs; Theadius of BCagnesia, who composed a 

^ly good book of Elements and generalised propositions, 

which had been confined to particular cases; Hermotimus of 

CtlapbfUi, who discovered many propositions of the Elements 

tad composed some on loci; and, finally, the names of Amyclas 

•f HeradMy Cyxicenas of Athens, and Philippus of Mende. 



34 A HISTOBY OF MATHEMATICS. 

A skilful mathematician of whose life and works we have 
no details is Aristasus^ the elder, probably a senior contempo- 
rary of Euclid. The fact that he wrote a work on conic 
sections tends to show that much progress had been made in 
their study during the time of Menaechmus. Aristseius wrote 
also on regular solids and cultivated *the analytic method. 
His works contained probably a summary of the researches 
of the Platonic school.* 

Aristotle (384-322 b.c), the systematiser of deductive logic, 
though not a professed mathematician, promoted the science 
of geometry by improving some of the most difficult defini- 
tions. His Physics contains passages with suggestive hints 
of the principle of virtual velocities. About his time there 
appeared a work called Me(^anica, of which he is regarded 
by some as the author. Mechanics was totally neglected by 
the Platonic school. 

The First Alexandrian School, 

In the previous pages we have seen the birth of geometry 
in Egypt, its transference to the Ionian Islands, thence to 
Lower Italy and to Athens. We have witnessed its growth 
in Greece from feeble childhood to vigorous manhood, and 
now we shall see it return to the land of its birth and there 
derive new vigour. 

During her declining years, immediately following the 
Peloponnesian War, Athens produced the greatest scientists 
and philosophers of antiquity. It was the time of Plato 
and Aristotle. In 338 b.c, at the battle of Chseronea, Athens 
was beaten by Philip of Macedon, and her power was broken 
forever. Soon after, Alexander the Great, the son of Philip, 
started out to conquer the world. In eleven years he built 
up a great empire which broke to pieces in a day. Egypt 



THE GREEKS. 85 

Cell to the lot of Ptolemy Soter. Alexander had founded 
the seaport of Alexandria, which soon became 'Hhe noblest 
of an cities." Ptolemy made Alexandria the capital. The 
histoiy of Egypt during the next three centuries is mainly 
the history of Alexandria. Literature, philosophy, and art 
were diligently cultivated. Ptolemy created the university 
of Alexandria. He founded the great Library and built labo- 
ntories, museums, a zoological garden, and promenades. Alex- 
andria soon became the great centre of learning. 

Demetrius Pbalereus was invited from Athens to take 
darge of the Library, and it is probable, says Gow, that 
Iidid was invited with him to open the mathematical school. 
Eadid's greatest activity was during the time of the first 
Ptolemy, who reigned from 306 to 283 b.c. Of the life of 
£aclid, little is known, except what is added by Proclus to 
the Eudemian Summary, Euclid, says Proclus, was younger 
than Plato and older than Eratosthenes and Archimedes, the 
latter of whom mentions him. He was of the Platonic sect, and 
▼ell read in its doctrines. He collected the Elements, put 
io Older much that Eudoxus had prepared, completed many 
tluDgs of Thesetetus, and was the first who reduced to unob- 
J^ctkmable demonstration the imperfect attempts of his prede- 
CttJors. When Ptolemy once asked him if geometry could 
w* be mastered by an easier process than by studying the 
BtmenUy Euclid returned the answer, "There is no royal 
wri to geometry." Pappus states that Euclid was distin- 
K^bed by the fairness and kindness of his disposition, par- 
**c^ly toward those who could do anything to advance 
ue mathematical sciences. Pappus is evidently making a 
contrast to Apollonius, of whom he more than insinuates the 
^*P|»eite character.' A pretty little story is related by Sto- 
**08:* "A youth who had begun to read geometry with 
Soelid, when he had learnt the first proposition, inquired, 



86 A HISTORY OF MATHEMATICS. 

' What do I get by learning these things ? ' So Euclid oalled 
his slave and said, 'Give him threepence, since be must 
make gain out of what he leams.' " These are about all the 
personal details preserved by Greek writers. Syrian and 
Arabian writers claim to know much more, but they are unre- 
liable. At one time Euclid of Alexandria was universally 
confounded with Euclid of Megara, who lived a century 
earlier. 

The fame of Euclid has at all times rested mainly upon his 
book on geometry, called the Elements, This book was so far 
superior to the Elements written by Hippocrates, Leon, and 
Theudius, that the latter works soon perished in the struggle 
for existence. The Greeks gave Euclid the special title of 
'< the author of the Elements.'^ It is a remarkable fact in the 
history of geometry, that the Elements of Euclid, written two 
thoiusand years ago, are still regarded by many as the best 
introduction to the mathematical sciences. In England they 
are used at the present time extensively as a text-book in 
schools. Some editors of Euclid have, however, been inclined 
to credit him with more than is his due. They would have 
us believe that a finished and unassailable system of geometry 
sprang at once from the brain of Euclid, " an armed Minerva 
from the head of Jupiter." They fail to mention the earlier 
eminent mathematicians from whom Euclid got his material 
Comparatively few of the propositions and proofs in the 
Elements are his own discoveries. In fact, the proof of the 
" Theorem of Pythagoras " is the only one directly ascribed to 
him. Allman conjectures that the substance of Books I., II., 
IV. comes from the Pythagoreans, that the substance of Book 
VI. is due to the Pythagoreans and Eudoxus, the latter con- 
tributing the doctrine of proportion as applicable to incom- 
mensurables and also the Method of Exhaustions (Book VII.), 
that Thesetetus contributed much toward Books X. and XIIL, 



THE GREEKS. 37 

that the principal paxt of the original work of Euclid himself 
U to be found in Book X.^ Euclid was the greatest systema- 
tiser of his time. By careful selection from the material 
before him, and by logical arrangement of the propositions 
selected, he built up, from a few definitions and axioms, a 
piood and lofty structure. It would be erroneous to believe 
tbat he incorporated into his Elements all the elementary 
theorems known at his time. Archimedes, Apollonius, and 
eren he himself refer to theorems not included in his Ele- 
metUij as being well-known truths. 

The text of the Elements now commonly used is Theon's 
edition. Theon of Alexandria, the father of Hypatia, brought 
oat an edition, about 700 years after Euclid, with some altera- 
tions in the text. As a consequence, later commentators, 
especially Robert Simson, who laboured under the idea that 
Euclid must be absolutely perfect, made Theon the scape- 
goat for all the defects which they thought they could discover 
in the text as they knew it. But among the manuscripts sent 
by Napoleon I. from the Vatican to Paris was found a copy of 
the Elements believed to be anterior to Theon's recension. 
ILiny variations from Theon's version were noticed therein, 
bat they were not at all important, and showed that Theon 
generally made only verbal changes. The defects in the 
Elements for which Theon was blamed must, therefore, be 
due to Euclid himself. The Elements has been considered as 
offering models of scrupulously rigorous demonstrations. It 
is certainly true that in point of rigour it compares favourably 
with its modem rivals; but when examined in the light of 
strict mathematical logic, it has been pronounced by C. S. 
Peirce to be " riddled with fallacies." The results are correct 
only because the writer's experience keeps him on his guard. 

At the beginning of our editions of the Elements, under 
the head of definitions, are given the assumptions of such 



88 A HISTORY OF MATHEMATICS. 

notions as the point, line, etc., and some verbal explanations. 
Then follow three postulates or demands, and twelve axioms. 
The term 'axiom' was used by Proclus, but not by Euclid. 
He speaks, instead, of 'common notions' — common either 
to all men or to all sciences. There has been much contro- 
versy among ancient and modern critics on the postulates and 
axioms. An immense preponderance of manuscripts and the 
testimony of Proclius place the 'axioms' about right angles 
and parallels (Axioms 11 and 12) among the postulates.^'" 
This is indeed their proper place, for they are really assump- 
turns, and not common notions or axioms. The postulate 
about parallels plays an important rdle in the history of non- 
Euclidean geometry. The only postulate which Euclid missed 
was the one of superposition, according to which figures 
can be moved about in space without any alteration in form 
or magnitude. 

The Elements contains thirteen books by Euclid, and two^ 
of which it is supposed that Hypsicles and Damascius are 
the authors. The first four books are on plane geometry. 
The fifth book treats of the theory of proportion as applied 
to magnitudes in general. The sixth book develops the 
geometry of similar figures. The seventh, eighth, ninth 
books are on the theory of numbers, or on arithmetic. In the 
ninth book is found the proof to the theorem that the number 
of primes is infinite. The tenth book treats of the theory of 
incommensurables. The next three books are on stereometry. 
The eleventh contains its more elementary theorems; the 
twelfth, the metrical relations of the pyramid, prism, cone, 
cylinder, and sphere. The thirteenth treats of the regular 
polygons, especially of the triangle and pentagon, and then uses 
them as faces of the five regular solids ; namely, the tetraedron, 
octaedron, icosaedron, cube, and dodecaedron. The regular 
solids were studied so extensively by the Platonists that they 



THE 6BEEKS. 89 

leoeived the name of ''Platonic figures." The statement of 
Produs that the whole aim of Euclid in writing the Elements 
was to arrive at the construction of the regular solids, is 
obyiooslj wrong. The fourteenth and fifteenth books, treat- 
ing of solid geometry, are apocryphal. 

A remarkable feature of Euclid's, and of all Greek geometry 
before Archimedes is that it eschews mensuration. Thus the 
theorem that the area of a triangle equals half the product 
of its base and its altitude is foreign to Euclid. 

Another extant book of Euclid is the Data, It seems to 
bare been written for those who, having completed the Ele- 
ments, wish to acquire the power of solving new problems 
proposed to them. The Data is a course of practice in analy- 
iML It contains little or nothing that an intelligent student 
eoald not pick up from the Elements itself. Hence it contrib- 
utes little to the stock of scientific knowledge. The following 
ire the other extant works generally attributed to Euclid: 
JPkcenamena, a work on spherical geometry and astronomy; 
OpiicSy which develops the hypothesis that light proceeds 
from the eye, and not from the object seen ; Catoptricay con- 
taining propositions on reflections from mirrors ; De Divisioni- 
lm9j a treatise on the division of plane figures into parts 
having to one another a given ratio ; Sectio Canonis, a work 
on musical intervals. His treatise on Porisms is lost; but 
much learning has been expended by Eobert Simson and 
M. Cbasles in restoring it from numerous notes found in the 
writings of Pappus. The term 'porism ' is vague in meaning. 
The aim of a porism is not to state some property or truth, 
like a theorem, nor to effect a construction, like a problem, 
but to find and bring to view a thing which necessarily exists 
with given numbers or a given construction, as, to find the 
eentre of a given circle, or to find the G.C.D. of two given 
numbers.* His other lost works are FaUadeSy containing 



40 A HISTORY OF MATHEliATICS. ' 

exercises in detection of fallacies; Conic Sections, in four 
books, which are the foundation of a work on the same sub- 
ject by Apollonius; and Loci on a Surface, the meaning of 
which title is not understood. Heiberg believes it to mean 
"loci which are surfaces." 

The immediate successors of Euclid in the mathematical 
school at Alexandria were probably Conon, Dositheus, and 
ZenxippUB, but little is known of them. 

Archimedes (287 ?-212 b.c), the greatest mathematician of 
antiquity, was born in Syracuse. Plutarch calls him a rela- 
tion of King Hieron; but more reliable is the statement of 
Cicero, who tells us he was of low birth. Diodorus says he 
visited Egypt, and, since he was a great friend of Conon and 
Eratosthenes, it is highly probable that he studied in Alexan- 
dria. This belief is strengthened by the fact that he had 
the most thorough acquaintance with all the work previously 
done in mathematics. He returned, however, to Syracuse, 
where he made himself useful to his admiring friend and 
patron. King Hieron, by applying his extraordinary inventive 
genius to the construction of various war-engines, by which 
he inflicted much loss on the Romans during the siege of 
Marcellus. The story that, by the use of mirrors reflecting 
the sun's rays, he set on Are the Roman ships, when they 
came within bow-shot of the walls, is probably a fiction. The 
city was taken at length by the Romans, and Archimedes 
perished in the indiscriminate slaughter which followed. Ac- 
cording to tradition, he was, at the time, studying the diagram 
to some problem drawn in the sand. As a Roman soldier 
approached him, he called out, "Don't spoil my circles." 
The soldier, feeling insulted, rushed upon him and killed 
him. No blame attaches to the Roman general Marcellus, 
who admired his genius, and raised in his honour a tomb 
bearing the figure of a sphere inscribed in a cylinder. When 



THB GBBBKS. 41 

Cieero was in Syracuse, he found the tomb buried under 
mblHsh. 

Aichimedes was admired by his fellow-citizens chiefly for 

his mechanical inventions ; he himself prized far more highly 

his discoveries in pure science. He declared that '< every kind 

of art which was connected with daily needs was ignoble and 

Tulgar." Some of his works have been lost. The following 

aie the extant books, arranged approximately in chronological 

order : 1. Two books on Equiponderance of Planes or Centres 

of Plane OravUies, between which is inserted his treatise on 

the Quadrature of the Parabola; 2. Two books on the Sphere 

UkACylinder; 3. The Measurement of the Circle ; 4. On Spirals; 

& Conoids and Spheroids; 6. The Sand-Counter; 7. Two books 

OD Floating Bodies; 8. Fifteen Lemmas, 

In the book on the Measurement of the Circle, Archimedes 
proves first that the area of a circle is equal to that of a 
right triangle having the length of the circumference for its 
base, and the radius for its altitude. In this he assumes that 
there exists a straight line equal in length to the circumference 
— an assumption objected to by some ancient critics, on 
the ground that it is not evident that a straight line can equal 
a curved one. The finding of such a line was the next prob- 
lem. He first finds an upper limit to the ratio of the circum- 
ference to the diameter, or ir. To do this, he starts with an 
equilateral triangle of which the base is a tangent and the 
vertex is the centre of the circle. By successively bisecting 
the angle at the centre, by comparing ratios, and by taking the 
irrational square roots always a little too small, he finally 
arrived at the conclusion that ir < 3|. Next he finds a lower 
limit by inscribing in the circle regular polygons of 6, 12, 24, 
48y 96 sides, finding for each successive polygon its perimeter, 
which is, of course, always less than the circumference. Thus 
li6 finaUy concludes that ''the circumference of a circle ex- 



42 A HISTOBY OF MATHBMATIGS. 

ceeds three times its diameter by a part which is less than ^ 
but more than ^^ of the diameter." This approximation is 
exact enough for most purposes. 

The Quadrature of the Parabola contains two solutions to 
the problem — one mechanical, the other geometrical. The 
method of exhaustion is used in both. 

Archimedes studied also the ellipse and accomplished its 
quadrature, but to the hyperbola he seems to have paid less at- 
tention. It is believed that he wrote a book on conic sections. 

Of all his discoveries Archimedes prized most highly those 
in his Sphere and Cylinder. In it are proved the new 
theorems, that the surface of a sphere is equal to four times 
a great circle ; that the surface of a segment of a sphere is 
equal to a circle whose radius is the straight line drawn from 
the vertex of the segment to the circumference of its basal 
circle ; that the volume and the surface of a sphere are \ of 
the volume and surface, respectively, of the cylinder circum- 
scribed about the sphere. Archimedes desired that the figure 
to the last proposition be inscribed on his tomb. This was 
ordered done by Marcellus. 

The spiral now called the "spiral of Archimedes," and 
described in the book On Spirals, was discovered by Archi- 
medes, and not, as some believe, by his friend Conon.' His 
treatise thereon is, perhaps, the most wonderful of all his 
works. Nowadays, subjects of this kind are made easy by 
the use of the infinitesimal calculus. In its stead the ancients 
used the method of exhaustion. Nowhere is the fertility of 
his genius more grandly displayed than in his masterly use of 
this method. With Euclid and his predecessors the method 
of exhaustion was only the means of proving propositions 
which must have been seen and believed before they were 
proved. But in the hands of Archimedes it became an instru- 
ment of discovery.* 



THE GREEKS. 48 

By the word 'conoid/ in his book on Conoids and 
SpkertridSy is meant the solid produced by the revolution 
of a parabola or a hyperbola about its axis. Spheroids 
aie produced by the revolution of an ellipse, and are long 
or flat, according as the ellipse revolves around the major 
or minor axis. The book leads up to the cubature of these 
•olidB. 

We have now reviewed briefly all his extant works on geom- 
etry. His arithmetical treatise and problems will be consid- 
ered later. We shall now notice his works on mechanics. 
Archimedes is the author of the first sound knowledge on this 
nbject ArchytaSy Aristotle, and others attempted to form 
the known mechanical truths into a science, but failed. Aris- 
totle knew the property of the lever, but could not establish 
its true mathematical theory. The radical and fatal defect 
in the speculations of the Greeks, says Whewell, was ''that 
though they had in their possession facts and ideas, the ideas 
were not distinct and appropriate to the facts." For instance, 
Aristotle asserted that when a body at the end of a lever is 
moving, it may be considered as having two motions ; one in 
the direction of the tangent and one in the direction of the 
radius ; the former motion is, he says, according to nature, the 
latter contrary to nature. These inappropriate notions of 
'natural' and 'unnatural' motions, together with the habits 
of thought which dictated these speculations, made the per- 
ception of the true grounds of mechanical properties impos- 
sible." It seems strange that even after Archimedes had 
entered upon the right path, this science should have remained 
absolutely stationary till the time of Galileo — a period of 
nearly two thousand years. 

The proof of the property of the lever, given in his Equi- 
pomderance of Planes, holds its place in text-books to this day. 
HiB estimate of the efficiency of the lever is expressed in the 



44 A HISTORY OF MATHEMATICS. 

saying attributed to him, ''Give me a fulcrum on which to 
rest, and I will move the earth." 

While the Equiponderance treats of solids, or the equilib- 
rium of solids, the book on Floating Bodies treats of hydro- 
statics. His attention was first drawn to the subject of 
specific gravity when King Hieron asked him to test whether 
a crown, professed by the maker to be pure gold, was not 
alloyed with silver. The story goes that our philosopher was 
in a bath when the true method of solution flashed on his 
mind. He immediately ran home, naked, shouting, '' I have 
found it ! " To solve the problem, he took a piece of gold and 
a piece of silver, each weighing the same as the crown. Ac- 
cording to one author, he determined the volume of water 
displaced by the gold, silver, and crown respectively, and 
calculated from that the amount of gold and silver in the 
crown. According to another writer, he weighed separately 
the gold, silver, and crown, while immersed in water, thereby 
determining their loss of weight in water. From these data 
he easily found the solution. It is possible that Archimedes 
solved the problem by both methods. 

After examining the writings of Archimedes, one can well 
understand how, in ancient times, an ' Archimedean problem ' 
came to mean a problem too deep for ordinary minds to solve, 
and how an 'Archimedean proof came to be the synonym for 
unquestionable certainty, Archimedes wrote on a very wide 
range of subjects, and displayed great profundity in each. He 
is the Newton of antiquity. 

Bratosthenes, eleven years younger than Archimedes, was a 
native of Cyrene. He was educated in Alexandria under 
Callimachus the poet, whom he succeeded as custodian of 
the Alexandrian Library. His many-sided activity may be 
inferred from his works. He wrote on Good and Evil, Meas- 
urement of the Earth, Comedy, Geography, Chronology, Constd- 



THE GREEKS. 45 

hUoiUy and the Duplication of the Cube. He was also a 
phUologian and a poet. He measured the obliquity of the 
ecliptic and invented a device for finding prime numbers. 
Of his geometrical writings we possess only a letter to 
Ptolemy Euergetes, giving a history of the duplication prob- 
lem and also the description of a very ingenious mechanical 
eoDtriyance of his own to solve it. In his old age he lost 
his eyesight, and on that account is said to have committed 
suicide by voluntary starvation. 

About forty years after Archimedes flourished ApoUonius of 
Ferga, whose genius nearly equalled that of his great prede- 
cessor. He incontestably occupies the second place in dis- 
tinction among ancient mathematicians. ApoUonius was 
bom in the reign of Ptolemy Euergetes and died under 
Ptolemy Philopator, who reigned 222-206 b.c. He studied at 
Akzandria tinder the successors of Euclid, and for some time, 
tlao, at Pergamum, where he made the acquaintance of that 
Eodemus to whom he dedicated the first three books of his 
Conic Sections, The brilliancy of his great work brought him 
the title of the " Great Geometer." This is all that is known 
of his life. 

His Conic Sections were in eight books, of which the first 
foor only have come down to us in the original Greek. The 
next three books were unknown in Europe till the middle of 
the seventeenth century, when an Arabic translation, made 
about 1250, was discovered. The eighth book has never been 
found. In 1710 Halley of Oxford published the Greek text 
of the first four books and a Latin translation of the remain- 
ing three, together with his conjectural restoration of the 
eighth book, founded on the introductory lemmas of Pappus. 
The first four books contain little more than the substance 
of what earlier geometers had done. Eutocius tells us that 
Heraclides, in his life of Archimedes, accused ApoUonius of 



46 A HISTORY OF MATHBMATIC8. 

having appropriated, in his Conic Sections, the unpublished 
discoveries of that great mathematician. It is difficult to 
believe that this charge rests upon good foundation. Eutocius 
quotes Geminus as replying that neither Archimedes nor 
Apollonius claimed to have invented the conic sections^ but 
that Apollonius had introduced a real improvement. While 
the first three or four books were founded on the works of 
MensechmuSy Aristseus, Euclid, and Archimedes, the remaining 
ones consisted almost entirely of new matter. The first three 
books were sent to Eudemus at intervals, the other books 
(after Eudemus's death) to one Attains. The preface of the 
second book is interesting as showing the mode in which 
Greek books were ^published' at this time. It reads thus: 
'^ I have sent my son Apollonius to bring you (Eudemus) the 
second book of my Conies. Bead it carefully and communi- 
cate it to such others as are worthy of it. If Philonides, the 
geometer, whom I introduced to you at Ephesus, comes into 
the neighbourhood of Pergamum, give it to him also." " 

The first book, says Apollonius in his preface to it, "con- 
tains the mode of producing the three sections and the conju- 
gate hyperbolas and their principal characteristics, more fully 
and generally worked out than in the writings of other 
authors." We remember that Menaechmus, and all his suc- 
cessors down to Apollonius, considered only sections of right 
cones by a plane perpendicular to their sides, and that the 
three sections were obtained each from a different cone. 
Apollonius introduced an important generalisation. He pro- 
duced all the sections from one and the same cone, whether 
right or scalene, and by sections which may or may not be 
perpendicular to its sides. The old names for the three curves 
were now no longer applicable. Instead of calling the three 
curves, sections of the * acute-angled,' 'right-angled,' and 
'obtuse-angled' cone, he called them ellipse, parabola, and 



THE GBEEKS. 47 

knperMa, respectively. To be sure, we find the words ' parab- 
ola 'and 'ellipse' in the works of Archimedes, but they are 
{ffobably only interpolations. The word 'ellipse ' was applied 
^'c^ii^^i^ t^<P^P being the parameter; the word 'parabola' 
was introduced because y*=:px, and the term 'hyperbola' 
because y^>px. 

The treatise of Apollonius rests on a unique property of 
ctmic sections, which is derived directly from the nature of 
the cone in which these sections are found. How this property 
forms the key to the system of the ancients is told in a mas- 
terly way by M. Chasles.^ "Conceive," says he, "an oblique 
eoDe on a circular base ; the straight line drawn from its 
Bummit to the centre of the circle forming its base is called 
the axis of the cone. The plane passing through the axis, 
perpendicular to its base, cuts the cone along two lines and 
determines in the circle a diameter ; the triangle having this 
diameter for its base and the two lines for its sides, is called 
ike triangle through the axis. In the formation of his conic 
sections, Apollonius supposed the cutting plane to be perpen- 
dicular to the plane of the triangle through the axis. The 
points in which this plane meets the two sides of this triangle 
are the vertices of the curve ; and the straight line which joins 
these two points is a diameter of it. Apollonius called this 
diameter laius transversum. At one of the two vertices of the 
curve erect a perpendicular (latus rectum) to the plane of the 
triangle through the axis, of a certain length, to be determined 
as we shall specify later, and from the extremity of this per- 
pendicular draw a straight line to the other vertex of the 
curve ; now, through any point whatever of the diameter of 
the curve, draw at right angles an ordinate : the square of this 
ordinate, comprehended between the diameter and the curve, 
will be equal to the rectangle constructed on the portion of 
the ordinate comprised between the diameter and the straight 



48 A HISTOBY OF MATHEMATI08. 

line, and the part of the diameter comprised between the first 
vertex and the foot of the ordinate. Such is the charactenstio 
property which Apollonius recognises in his conic sections and 
which he uses for the purpose of inferring from it, by adroit 
transformations and deductions, nearly all the rest. It plays, 
as we shall see, in his hands, almost the same rdle as the 
equation of the second degree with two variables (abscissa and 
ordinate) in the system of analytic geometry of Descartes. 

'<It will be observed from this that the diameter of the 
curve and the perpendicular erected at one of its extremities 
suffice to construct the curve. These are the two elements 
which the ancients used, with which to establish their theory 
of conies. The perpendicular in question was called by them 
latu8 erectum; the moderns changed this name first to that of 
latiLS rectum, and afterwards to that of parameter J' 

The first book of the Conic Sections of Apollonius is almost 
wholly devoted to the generation of the three principal conic 
sections. 

The second book treats mainly of asymptotes, axes, and 
diameters. 

The third book treats of the equality or proportionality 
of triangles, rectangles, or squares, of which the component 
parts are determined by portions of transversals, chords, 
asymptotes, or tangents, which are frequently subject to a 
great number of conditions. It also touches the subject of 
foci of the ellipse and hyperbola. 

In the fourth book, Apollonius discusses the harmonic divis- 
ion of straight lines. He also examines a system of two 
conies, and shows that they cannot cut each other in more 
than four points. He investigates the various possible relative 
positions of two conies, as, for instance, when they have one 
or two points of contact with each other. 

The fifth book reveals better than any other the giant 



THE GREEKS. 49 

intellect of its author. Difficult questions of maxima and 
wUmma, of which few examples are found in earlier works, are 
beie treated most exhaustively. The subject investigated is, 
to find the longest and shortest lines that can be drawn from 
a given point to a conic. Here are also found the germs of 
the subject of evolutes and centres of osculation. 
The sixth book is on the similarity of conies. 
The seventh book is on conjugate diameters. 
The eighth book, as restored by Halley, continues the sub- 
ject of conjugate diameters. 

It is worthy of notice that Apollonius nowhere introduces 
the notion of directrix for a conic, and that, though he inciden- 
tally discovered the focita of an ellipse and hyperbola, he did 
not discover the focus of a parabola.^ Conspicuous in his 
geometry is also the absence of technical terms and symbols, 
which renders th^ proofs long and cumbrous. 

The discoveries of Archimedes and Apollonius, says M. 
Chasles,'* marked the most brilliant epoch of ancient geometry. 
Two questions which have occupied geometers of all periods 
may be regarded as having originated with them. The first 
of these is the quadrature of curvilinear figures, which gave 
birth to the infinitesimal calculus. The second is the theory 
of conic sections, which was the prelude to the theory of 
geometrical curves of all degrees, and to that portion of 
geometry which considers only the forms and situations 
of figures, and uses only the intersection of lines and surfaces 
and the ratios of rectilineal distances. These two great 
divisions of geometry may be designated by the names of 
Geometry of Measurements and Geometry of Forms and Situa- 
tions, or, Greometry of Archimedes and of Apollonius. 

Besides the Conic Sections, Pappus ascribes to Apollonius 
the following works: On Contacts, Plane Loci, Inclinations, 
Section of an Area, Determinate Section, and gives lemmas 



60 A HISTORY OF MATHBMATICS. 

from which attempts have been made to restore the lost , 
originals. Two books on De Sectume BcUumU have been 
found in the Arabic. The book on ContactSy as restored by 
Vieta, contains the so-called '< Apollonian Problem": Given 
three circles, to find a fourth which shall touch the three. 

Euclid, Archimedes, and Apollonius brought geometry to 
as high a state of perfection as it perhaps could be brought 
without first introducing some more general and more powerful 
method than the old method of exhaustion. A briefer sym- 
bolism, a Cartesian geometry, an infinitesimal calculus, were 
needed. The Greek mind was not adapted to the invention of 
general methods. Instead of a climb to still loftier heights 
we observe, therefore, on the part of later Greek geometers, a 
descent, during which they paused here and there to look 
around for details which had been passed by in the hasty 
ascent.' 

Among the earliest successors of Apollonius was Nicomedea. 
Nothing definite is known of him, except that he invented the 
conchoid (^'mussel-like"). He devised a little machine by 
which the curve could be easily described. With aid of the 
conchoid he duplicated the cube. The curve can also be used 
for trisecting angles in a way much resembling that in the 
eighth lemma of Archimedes. Proclus ascribes this mode of 
trisection to Nicomedes, but Pappus, on the other hand, claims 
it as his own. The conchoid was used by Newton in con- 
structing curves of the third degree. 

About the time of Nicomedes, flourished also Diocles, the 
inventor of the cissoid ("ivy-like"). This curve he used for 
finding two mean proportionals between two given straight 
lines. 

About the life of Perseus we know as little as about that of 
Nicomedes and Diocles. He lived some time between 200 and 
100 B.C. From Heron and Geminus we learn that he wrote a 



THE GREEKS. 61 

vork on the jpire, a sort of anchor-ring surface described by 
Heron as being produced by the revolution of a circle around 
one of its chords as an axis. The sections of this surface 
yield peculiar curves called spiral sectionSy which, according to 
Geminus, were thought out by Perseus. These curves appear 
to be the same as the Hippopede of Eudoxus. 

Probably somewhat later than Perseus lived Zenodoras. He 
wrote an interesting treatise on a new subject; namely, iso- 
penmetrical Jiffures. Fourteen propositions are preserved by 
Pjtppos and Theon. Here are a few of them : Of isoperimet- 
nod, regular polygons, the one having the largest number of 
u^ has the greatest area; the circle has a greater area than 
tny regular polygon of equal periphery ; of all isoperimetrical 
polygons of n sides, the regular is the greatest ; of all solids 
IttTing surfaces equal in area, the sphere has the greatest 
▼dome. 

lypddes (between 200 and 100 b.c.) was supposed to be 
the author of both the fourteenth and fifteenth books of 
Eodid, but recent critics are of opinion that the fifteenth 
book was written by an author who lived several centuries 
1^ Christ. The fourteenth book contains seven elegant 
theorems on regular solids. A treatise of Hypsicles on Risings 
i> of interest because it is the first Greek work giving the 
^TJsion of the circumference into 360 degrees after the fash- 
lOQ of the Babylonians. 

Hipparclms of Nicsea in Bithynia was the greatest astron- 
omer of antiquity. He established inductively the famous 
theory of epicycles and eccentrics. As might be expected, he 
^a« interested in mathematics, not per se, but only as an aid 
to astronomical inquiry. No mathematical writings of his 
*w extant, but Theon of Alexandria informs us that Hippar- 
^^ originated the science of trigonometrtfj and that he calcu- 
^•W a " table of chords " in twelve books. Such calculations 



52 A HISTORY OF MATHSHATIGS. 

must have required a ready knowledge of arithmetical and 
algebraical operations. 

About 155 B.C. flourished Heron the Blder of Alexandria. 
He was the pupil of Ctesibius, who was celebrated for his 
ingenious mechanical inventions, such as the hydraulic organ, 
the water-clock, and catapult. It is believed by some that 
Heron was a son of Ctesibius. He exhibited talent of the 
same order as did his master by the invention of the eolipile 
and a curious mechanism known as ''Heron's fountain." 
Great uncertainty exists concerning his writings. Most au- 
thorities believe him to be the author of an important TreaHse 
on the Dioptra, of which there exist three manuscript copies, 
quite dissimilar. But M. Marie ^^ thinks that the Dioptra is 
the work of Heron the Younger^ who lived in the seventh or 
eighth century after Christ, and that Oeodesy^ another book 
supposed to be by Heron, is only a corrupt and defective copy 
of the former work. Dioptra contains the important formula 
for finding the area of a triangle expressed in terms of its 
sides; its derivation is quite laborious and yet exceedingly 
ingenious. ^' It seems to me difficult to believe," says Chasles, 
'' that so beautiful a theorem should be found in a work so 
ancient as that of Heron the Elder, without that some Greek 
geometer should have thought to cite it." Marie lays great 
stress on this silence of the ancient writers, and argues from 
it that the true author must be Heron the Younger or some 
writer much more recent than Heron the Elder. But no reli- 
able evidence has been found that there actually existed a 
second mathematician by the name of Heron. 

"Dioptra," says Venturi, were instruments which had great 
resemblance to our modern theodolites. The book Dioptra is 
a treatise on geodesy containing solutions, with aid of these 
instruments, of a large number of questions in geometry, such 
as to find the distance between two points, of which one only 



THE GREEKS. 58 

is aeeessible, or between two points which are visible but both 
inaooessible ; from a given point to draw a perpendicular to a 
line which cannot be approached ; to find the difference of 
lerel between two points ; to measure the area of a field with- 
out entering it. 

Heron was a practical surveyor. This may account for the 
bet that his writings bear so little resemblance to those of 
the Greek authors, who considered it degrading the science 
to apply geometry to surveying. The character of his geom- 
etry is not Grecian, but decidedly Egyptian. This fact is the 
more surprising when we consider that Heron demonstrated 
Us fiuniliarity with Euclid by writing a commentary on the 
Skwkents/^ Some of Heron's formulas point to an old Egyp- 
tiin origin. Thus, besides the above exact formula for the 
uea of a triangle in terms of its sides, Heron gives the for- 

mnla ^"^^ x j^ which bears a striking likeness to the for- 

nmla ^^^^ x ^\^ * for finding the area of a quadrangle, 

found in the Edfu inscriptions. There are, moreover, points 
of resemblance between Heron's writings and the ancient 
Ahmes papyrus. Thus Ahmes used unit-fractions exclusively; 
Heron uses them oftener than other fractions. Like Ahmes 
tnd the priests at Edfu, Heron divides complicated figures 
into simpler ones by drawing auxiliary lines; like them, 
he shows, throughout, a special fondness for the isosceles 
trapezoid. 

The writings of Heron satisfied a practical want, and for 
that reason were borrowed extensively by other peoples. We 
find traces of them in Rome, in the Occident during the Middle 
Ages, and even in India. 

Geminus of Ehodes (about 70 b.c.) published an astronomi- 
cal work still extant. He wrote also a book, now lost, on the 
ArratkQement of MathematicSy which contained many valuable 



54 A HISTORY OF MATHEMATICS. 

notices of the early history of Greek mathematics. Proclus 
and Eutocius quote it frequently. Theodosius of Tripolis is 
the author of a book of little merit on the geometry of the 
sphere. Dionysodorus of Amisus in Pontus applied the inter- 
section of a parabola and hyperbola to the solution of a prob- 
lem which Archimedes, in his Sphere and Cylinder, had left 
incomplete. The problem is ^'to cut a sphere so that its seg- 
ments shall be in a given ratio." 

We have now sketched the progress of geometry down to 
the time of Christ. Unfortunately, very little is known of 
the history of geometry between the time of ApoUonius and 
the beginning of the Christian era. The names of quite a 
number of geometers have been mentioned, but very few of 
their works are now extant. It is certain, however, that there 
were no mathematicians of real genius from ApoUonius to 
Ptolemy, excepting Hipparchus and perhaps Heron, 

The Second Alexandrian School, 

The close of the dynasty of the Lagides which ruled Egypt 
from the time of Ptolemy Soter, the builder of Alexandria, 
for 300 years ; the absorption of Egypt into the Roman Em- 
pire ; the closer commercial relations between peoples of the 
East and of the West; the gradual decline of paganism and 
spread of Christianity, — these events were of far-reaching 
influence on the progress of the sciences, which then had their 
home in Alexandria. Alexandria became a commercial and 
intellectual emporium. Traders of all nations met in her 
busy streets, and in her magnificent Library, museums, lecture- 
halls, scholars from the East mingled with those of the 
West ; Greeks began to study older literatures and to com- 
pare them with their own. In consequence of this interchange 
of ideas the Greek philosophy became fused with Oriental 



TUB GBEEKS. 55 

philofiophj. Neo-Pythagoreanism and Neo-Platonism were the 
DuneB of the modified systems. These stood, for a time, in 
apposition to Christianity. The study of Platonism and 
Pythagorean mysticism led to the revival of the theory of 
numbers. Perhaps the dispersion of the Jews and their 
intfodnction to Greek learning helped in bringing about this 
lerivaL The theory of numbers became a favourite study. 
Hub new line of mathematical inq\iiry ushered in what we 
MMj call a new school. There is no doubt that even now 
geometry continued to be one of the most important studies 
in the Alexandrian course. This Second Alexandrian School 
may be said to begin with the Christian era. It was made 
bmoos by the names of Claudius Ptolemseus^ Diophantus, 
Poppas, Theon of Smyrna, Theon of Alexandria, lamblichus. 
Porphyrins, and others. 

By the side of these we may place Serenus of Antissa, as 
hiring been connected more or less with this new school. 
He wrote on sections of the cone and cylinder, in two books, 
one of which treated only of the triangular section of the cone 
through the apex. He solved the problem, "given a cone 
(cylinder), to find a cylinder (cone), so that the section of 
both by the same plane gives similar ellipses." Of particular 
interest is the following theorem, which is the foundation 
<rf the modem theory of 
^monies: If from D we 
draw Z>F, cutting the tri- 
*>igle ABC, and choose 
* on it, so that DE : DF 
== EB : UF, and if we draw 
tte line All, then every transversal through 2), such as 
^, will be divided by AH so that DK: DG== KJiJG. 
^fewUmt of Alexandria (about 98 a.d.) was the author of 
^cerica, a work extant in Hebrew and Arabic, but not 




56 A HISTORY OF MATHEMATICS. 

in Greek* In it he proves the theorems on the congmenoe 
of spherical triangles^ and describes their properties in 
much the same way as Euclid treats plane triangles. In 
it are also found the theorems that the sum of the three 
sides of a spherical triangle is less than a great circle^ and 
that the sum of the three angles exceeds two right angles. 
Celebrated are two theorems of his on plane and spherical 
triangles. The one on plane triangles is that, ''if the three 
sides be cut by a straight line, the product of the three seg- 
ments which have no common extremity is equal to the 
product of the other three.'' The illustrious Camot makes 
this proposition, known as the * lemma of Menelaus/ the base 
of his theory of transversals. The corresponding theorem for 
spherical triangles, the so-called ' regula sex quantitatum,' is 
obtained from the above by reading "chords of three seg- 
ments doubled," in place of " three segments." 

Claudias Ptolemsus, a celebrated astronomer, was a native of 
Egypt. Nothing is known of his personal history except that 
he flourished in Alexandria in 139 a.d. and that he made the 
earliest astronomical observations recorded in his works, in 
125 A.D., the latest in 151 a.d. The chief of his works are 
the Syntaxis Mathematica (or the Almagest, as the Arabs call 
it) and the Qeographica, both of which are extant. The 
former work is based partly on his own researches, but mainly 
on those of Hipparchus. Ptolemy seems to have been not 
so much of an independent investigator, as a corrector and 
improver of the work of his great predecessors. The Almagest 
forms the foundation of all astronomical science down to 
Copernicus. The fundamental idea of his system, the " Ptole- 
maic System," is that the earth is in the centre of the uni- 
verse, and that the sun and planets revolve around the earth. 
Ptolemy did considerable for mathematics. He created, for 
astronomical use, a trigonometry remarkably perfect in form. 



THB GREEKS. 57 

The foundation of this science was laid by the illustrious 
Hipparchns. 

The Jlfnagest is in 13 books. Chapter 9 of the first book 
shows how to calculate tables of chords. The circle is divided 
into 3G0 degrees, each of which is halved. The diameter is 
diTided into 120 divisions ; each of these into 60 parts, which ' 
are again subdivided into 60 smaller parts. In Latin, these 
parts were called partes miniUce primos and partes mintUce 
mcunda. Hence our names, 'minutes' and 'seconds.'* The 
sexagesimal method of dividing the circle is of Babylonian 
origin, and was known to Geminus and Hipparchus. But 
Ptolemy's method of calculating chords seems original with 
him. He first proved the proposition, now appended to 
Euclid VI. (Z>), that "the rectangle contained by the diag- 
onals of a quadrilateral figure inscribed in a circle is equal 
to both the rectangles contained by its opposite sides." He 
then shows how to find from the chords of two arcs the 
chords of their sum and difference, and from the chord of any 
SIC that of its half. These theorems he applied to the calcu- 
h^n of his tables of chords. The proofs of these theorems 
sre very pretty. 

Another chapter of the first book in the Almagest is devoted 
to trigonometry, and to spherical trigonometry in particular. 
Ptolemy proved the * lemma of Menelaus,' and also the ' regula 
•w quantitatum.' Upon these propositions he built up his 
trigonometry. The fundamental theorem of plane trigonome- 
tiy, that two sides of a triangle are to each other as the chords 
of double the arcs measuring the angles opposite the two 
•Wes, was not stated explicitly by him, but was contained 
unplicitly in other theorems. More complete are the proposi- 
tions in spherical trigonometry. 

The fact that trigonometry was cultivated not for its own 
*^ bat to aid astronomical inquiry, explains the rather 



58 A HISTORY OF MATHEMATIGS. 

startling fact that spherical trigonometry came to exist in a 
developed state earlier than plane trigonometry. 

The remaining books of the Almagest are on astronomy. 
Ptolemy has written other works which have little or no bear- 
ing on mathematics^ except one on geometry. Extracts frqm 
this book, made by Proclus, indicate that Ptolemy did not 
regard the parallelraxiom of Euclid as self-evident, and that 
Ptolemy was the first of the long line of geometers from 
ancient time down to our own who toiled in the vain attempt 
to prove it. 

Two prominent mathematicians of this time were Nicoma- 
chus and Theon of Smyrna. Their favourite study was theory 
of numbers. The investigations in this science culminated 
later in the algebra of Diophantus. But no important geom- 
eter appeared after Ptolemy for 160 years. The only occupant 
of this long gap was Sextus Julius Africanus, who wrote an 
unimportant work on geometry applied to the art of war, 
entitled Cestes, 

Pappus, probably born about 340 a.d., in Alexandria, was 
the last great mathematician of the Alexandrian school. His 
genius was inferior to that of Archimedes, Apollonius, and 
Euclid, who flourished over 600 years earlier. But living, 
as he did, at a period when interest in geometry was declin- 
ing, he towered above his contemporaries "like the peak 
of Teneriffa above the Atlantic." He is the author of a Com- 
mentai-y on the Almagest, a Commentary on Euclid^ s ElemerUs, 
a Commentary on the Analemma of Diodorus, — a writer of 
whom nothing is known. All these works are lost. Proclus, 
probably quoting from the Commentary on Eudidj says that 
Pappus objected to the statement that an angle equal to a 
right angle is always itself a right angle. 

The only work of Pappus still extant is his Mathemaiical 
Collections. This was originally in eight books, but the first 



THE GREEKS. 59 

and portions of the second are now missing. The MaJthemat- 
ieal CoUectiona seems to have been written by Pappus to supply 
the geometers of his time with a succinct analysis of the most 
diffictilt mathematical works and to facilitate the study of 
them by explanatory lemmas. But these lemmas are selected 
Teiy freely, and frequently have little or no connection with the 
tabject on hand. However, he gives very accurate summaries 
of the works of which he treats. The Mathematical Collections 
is invaluable to us on account of the rich information it 
gives on various treatises by the foremost Greek mathemati- 
oana, which are now lost. Mathematicians of the last century 
considered it possible to restore lost works from the risumi 
b^ Pappus alone. 

We shall now cite the more important of those theorems in 
the Mathematical Collections which are supposed to be original 
with Pappus. First of all ranks the elegant theorem re-dis- 
covered by Ghddin, over 1000 years later, that the volume 
generated by the revolution of a plane curve which lies wholly 
oa one side of the axis, equals the area of the curve multiplied 
ty the circumference described by its centre of gravity. 
I^appus proved also that the centre of gravity of a triangle is 
ftat of another triangle whose vertices lie upon the sides of 
^ first and divide its three sides in the same ratio. In the 
fourth book are new and brilliant propositions on the quadra- 
^ which indicate an intimate acquaintance with curved 
^ttrfaces. He generates the quadratrix as follows : Let a 
^iral line be drawn upon a right circular cylinder ; then the 
Perpendiculars to the axis of the cylinder drawn from each 
P^int of the spiral line form the surface of a screw. A plane 
P*«8€d through one of these perpendiculars, making any con- 
^ejiient angle with the base of the cylinder, cuts the screw- 
"^wface in a curve, the orthogonal projection of which upon 
^ base is the quadratrix, A second mode of generation is 



60 A HISTORY OF MATHEMATICS. 

no less admirable : If we make the spiral of Archimedes the 
base of a right cylinder, and imagine a cone of revolution 
having for its axis the side of the cylinder passing through 
the initial point of the spiral, then this cone cuts the cylinder 
in a curve of double curvature. The perpendiculars to the 
axis drawn through every point in this curve form the surface 
of a screw which Pappus here calls the plectoidal surface, A 
plane passed through one of the perpendiculars at any con- 
venient angle cuts that surface in a curve whose orthogonal 
projection upon the plane of the spiral is the required qtujutror 
trix. Pappus considers curves of double curvature still further. 
He produces a spherical spircU by a point moving uniformly 
along the circumference of a great circle of a sphere, while 
the great circle itself revolves uniformly around its diameter. 
He then finds the area of that portion of the surface of the 
sphere determined by the spherical spiral, ''a complanation 
which claims the more lively admiration, if we consider that, 
although the entire surface of the sphere was known since 
Archimedes' time, to measure portions thereof, such as spher- 
ical triangles, was then and for a long time afterwards an 
unsolved problem."* A question which was brought into 
prominence by Descartes and Newton is the "problem of 
Pappus." Given several straight lines in a plane, to find the 
locus of a point such that when perpendiculars (or, more 
generally, straight lines at given angles) are drawn from it to 
the given lines, the product of certain ones of them shall be in 
a giveii ratio to the product of the remaining ones. It is 
worth noticing that it was Pappus who first found the focus 
of the parabola, suggested the use of the directrix, and pro- 
pounded the theory of the involution of points. He solved 
the problem to draw through three points lying in the same 
straight line, three straight lines which shall form a triangle 
inscribed in a given circle.* From the McU?iem(Uic(il Collections 



THE GBBEKS. 61 

manj more equally difficult theorems might be quoted which 
are original with Pappus as far as we know. It ought to be 
lenuLrked, however, that he is known in three instances to 
IttTe copied tjieorems without giving due credit, and that he 
may have done the same thing in other cases in which we 
hare no data by which to ascertain the real discoverer. 

Aboat the time of Pappus lived Theon of Alexandria. He 
brought out an edition of Euclid's Elements with notes, which 
be probably used as a text-book in his classes. His commen- 
tary on the Almagest is valuable for the many historical notices, 
and especially for the specimens of Greek arithmetic which it 
contains. Theon's daughter Hypatia, a woman celebrated for 
ber beauty and modesty, was the last Alexandrian teacher of 
reputation, and is said to have been an abler philosopher and 
mathematician than her father. Her notes on the works of 
Diophantus and Apollonius have been lost. Her tragic death 
in 415 A.D. is vividly described in Kingsley's Hypatia, 

From now on, mathematics ceased to be cultivated in 

Alexandria. The leading subject of men's thoughts was 

Christian theology. Paganism disappeared, and with it pagan 

learning. The Neo-Platonic school at Athens struggled on a 

century longer. Proclus, Isidorus, and others kept up the 

^golden chain of Platonic succession." Produs, the successor 

of Syrianus, at the Athenian school, wrote a commentary on 

Euclid's Elements. We possess only that on the first book, 

which is valuable for the information it contains on the 

bistory of geometry. Damascias of Damascus, the pupil of 

hidoruB, is now believed to be the author of the fifteenth 

book of Euclid. Another pupil of Isidorus was Eutocius of 

Ascalon, the commentator of Apollonius and Archimedes. 

Simplicias wrote a commentary on Aristotle's De Codo, In 

the year 529, Justinian, disapproving heathen learning, finally 

closed by imperial edict the schools at Athens. 



62 A HISTOBT OF MATHEMATICS. 

As a rule^ the geometries of the last 500 years showed 
a lack of creative power. They were commentators rather 
than discoverers. 

The principal characteristics of ancient geometry are t — 

(1) A wonderful clearness and definiteness of its concepts 
and an almost perfect logical rigour of its conclusions. 

(2) A complete want of general principles and methods. 
Ancient geometry is decidedly special. Thus the Greeks 
possessed no general method of drawing tangents. ^^The 
determination of the tangents to the three conic sections did 
not furnish any rational assistance for drawing the tangent to 
any other new curve, such as the conchoid, the cissoid, etc." " 
In the demonstration of a theorem, there were, for the ancient 
geometers, as many different cases requiring separate proof 
as there were different positions for the lines. The greatest 
geometers considered it necessary to treat all possible cases 
independently of each other, and to prove each with equal 
fulness. To devise methods by which the various cases could 
all be disposed of by one stroke, was beyond the power of the 
ancients. ^'If we compare a mathematical problem with a 
huge rock, into the interior of which we desire to penetrate, 
then the work of the Greek mathematicians appears to us like 
that of a vigorous stonecutter who, with chisel and hammer, 
begins with indefatigable perseverance, from without, to 
crumble the rock slowly into fragments ; the modern mathe- 
matician appears like an excellent miner, who first bores 
through the rock some few passages, from which he then bursts 
it into pieces with one powerful blast, and brings to light the 
treasures within." " 



THE GREEKS. 68 

GREEK ARITHMETIC. 

Greek mathematicians were in the habit of discriminating 
between the science of numbers and the art of calculation. 
The former they called arithmetica, the latter logistica. The 
drawing of this distinction between the two was very natural 
and proper. The difference between them is as marked as 
that between theory and practice. Among the Sophists the 
art of calculation was a favourite study. Plato, on the other 
handy gave considerable attention to philosophical arithmetic^ 
bat pronounced calculation a vulgar and childish art. 

In sketching the history of Greek calculation, we shall first 
give a brief account of the Greek mode of counting and of 
writing numbers. Like the Egyptians and Eastern nations, 
the earliest Greeks counted on their fingers or with pebbles. 
In case of large numbers, the pebbles were probably ar- 
ranged in parallel vertical lines. Pebbles on the first line 
represented units, those on the second tens, those on the third 
hundreds, and so on. Later, frames came into use, in which 
strings or wires took the place of lines. According to tra- 
dition, Pythagoras, who travelled in Egypt and, perhaps, in 
India, first introduced this valuable instrument into Greece. 
The abacus, as it is called, existed among different peoples and 
at different times, in various stages of perfection. An abacus 
is still employed by the Chinese under the name of Swan-pan, 
We possess no specific information as to how the Greek abacus 
looked or how it was used. Boethius says that the Pytha- 
goreans used with the abacus certain nine signs called apices, 
which resembled in form the nine "Arabic numerals." But 
the correctness of this assertion is subject to grave doubts. 

The oldest Grecian numerical symbols were the so-called 
Herodianic signs (after Herodianus, a Byzantine grammarian of 
about 200 A.D., who describes them). These signs occur fre- 



64 A HISTORY OF MATHEMATICS. 

quently in Athenian inscriptions and are, on that account^ now 
generally called Attic. For some unknown reason these sym- 
bols were afterwards replaced by the alphabetic numerals^ in 
which the letters of the Greek alphabet were used^ together 
with three strange and antique letters ^, 9 > and 9, and the 
symbol M. This change was decidedly for the worse^ for the 
old Attic numerals were less burdensome on the memory, inas- 
much as they contained fewer symbols and were better adapted 
to show forth analogies in numerical operations. The follow- 
ing table shows the Greek alphabetic numerals and their 
respective values : — 

128456780 10 2OdOiO6O0O 70 8000 

100 200 300 400 600 600 700 800 900 1000 2000 8000 

^ y 

M M M etc. 

10,000 20,000 30,000 

It will be noticed that at 1000, the alphabet is begun over 
again, but, to prevent confusion, a stroke is now placed before 
the letter and generally somewhat below it. A horizontal line 
drawn over a number served to distinguish it more readily 
from words. The coefficient for M was sometimes placed 
before or behind instead of over the M. Thus 43,678 was 
written SM^yxorj, It is to be observed that the Greeks had no 
zero. 

Fractions were denoted by first writing the numerator 
marked with an accent, then the denominator marked with 
two accents and written twice. Thus, ly' kB"kB" = ^« In case 
of fractions having unity for the numerator, the a' was omitted 
and the denominator was written only once. Thus fiS'' = ^- 



i^ 


2 6 6 




i^ 


2 6 5 




MMfi,a 


40000, 12000, 


1000 


^fi.yk'r 


12000, 3600, 


300 


fii r KC 


1000, 300, 


25 


f 







M aicc 70225 



THB GSEBKS. 65 

Greek writers seldom refer to calculation with alphabetic 
Bimierals. Addition^ subtraction, and even multiplication were 
probably performed on the abacus. Expert mathematicians 
maj have used the symbols. Thus Eutocius, a commentator 
of the sixth century after Christ, gives a great many multipli- 
eations of which the following is a specimen : * — 

The operation is ex- 
plained sufficiently by the 
modern numerals append- 
ed. In case of mixed 
numbers, the process was 
still more clumsy. Divis- 
ions are found in Theon 
of Alexandria's commen- 
tary on the Almagest. As 
might be expected, the process is long and tedious. 

We have seen in geometry that the more advanced mathe- 
maticians frequently had occasion to extract the square root. 
Thus Archimedes in his Mensuration of the Cirde gives a 
large number of square roots. He states, for instance, that 
V3 < ^fj^ and V3 > fj|, but he gives no clue to the method 
by which he obtained these approximations. It is not im- 
probable that the earlier Greek mathematicians found the 
square root by trial only. Eutocius says that the method of 
extracting it was given by Heron, Pappus, Theon, and other 
commentators on the Almagest Theon's is the only ancient 
method known to us. It is the same as the one used nowa- 
days, except that sexagesimal fractions are employed in place 
of our decimals. What the mode of procedure actually was 
when sexagesimal fractions were not used, has been the sub- 
ject of conjecture on the part of numerous modern writers.*^ 

Of interest, in connection with arithmetical symbolism, is 
the SandrCounter (Arenarius), an essay addressed by Archi- 



66 A HISTOBY OF MATHEMATICS. 

medes to G^lon, king of Syracuse. In it Archimedes shows 
that people are in error who think the sand cannot be counted, 
or that if it can be counted^ the number cannot be expressed 
by arithmetical symbols. He shows that the number of grains 
in a heap of sand not only as large as the whole earthy but as 
large as the entire universe^ can be arithmetically expressed. 
Assuming that 10^000 grains of sand suffice to make a little 
solid of the magnitude of a poppy-seed^ and that the diameter 
of a poppy-seed be not smaller than ^ part of a finger's 
breadth ; assuming further, that the diameter of the universe 
(supposed to extend to the sun) be less than 10,000 diameters 
of the earth, and that the latter be less than 1,000,000 stadia^ 
Archimedes finds a number which would exceed the number 
of grains of sand in the sphere of the universe. He goes on 
even further. Supposing the universe to reach out to the fixed 
stars, he finds that the sphere, having the distance from the 
earth's centre to the fixed stars for its radius, would contain 
a number of grains of sand less than 1000 myriads of the 
eighth octad. In our notation, this number would be 10" or 
1 with 63 ciphers after it. It can hardly be doubted that one 
object which Archimedes had in view in making this calcula- 
tion was the improvement of the Greek symbolism. It is not 
known whether he invented some short notation by which to 
represent the above number or not. 

We judge from fragments in the second book of Pappus that 
ApoUonius proposed an improvement in the Greek method of 
writing numbers, but its nature we do not know. Thus we 
see that the Greeks never possessed the boon of a clear, com- 
prehensive symbolism. The honour of giving such to the world, 
once for all, was reserved by the irony of fate for a nameless 
Indian of an unknown time, and we know not whom to thank 
for an invention of such importance to the general progress of 
intelligence.* 



THE GREEKS. 67 

Ptosing from the subject of logistica to that of arithmetical 
our attention is first drawn to the science of numbers of 
Fftfaagoras. Before founding his school^ Pythagoras studied 
for many years under the Egyptian priests and familiarised 
himself with Egyptian mathematics and mysticism. If he 
ever was in Babylon, as some authorities claim, he may have 
learned the sexagesimal notation in use there ; he may have 
{Hcked up considerable knowledge on the theory of proportion, 
and may have found a large number of interesting astronomical 
observations. Saturated with that speculative spirit then 
pervading the Greek mind, he endeavoured to discover some 
principle of homogeneity in the universe. Before him, the 
philosophers of the Ionic school had sought it in the matter 
of things ; Pythagoras looked for it in the structure of things. 
He observed various numerical relations or analogies between 
numbers and the phenomena of the universe. Being convinced 
that it was in numbers and their relations that he was to find 
the foundation to true philosophy, he proceeded to trace the 
origin of all things to numbers. Thus he observed that musi- 
cal strings of equal length stretched by weights having the 
proportion of \, J, f , produced intervals which were an octave, 
a fifth, and a fourth. Harmony, therefore, depends on musi- 
cal proportion ; it is nothing but a mysterious numerical rela- 
tion. Where harmony is, there are numbers. Hence the 
order and beauty of the universe have their origin in numbers. 
There are seven intervals in the musical scale, and also seven 
planets crossing the heavens. The same mimerical relations 
which underlie the former must underlie the latter. But where 
numbers are, there is harmony. Hence his spiritual ear dis- 
cerned in the planetary motions a wonderful * harmony of the 
spheres.' The Pythagoreans invested particular numbers with 
extraordinary attributes. Thus one is the essence of things ; 
it is an absolute number ; hence the origin of all numbers and 



68 A HISTORY OF MATHEMATICS. 

SO of all things. Four is the most perfect number, and was in 
some mystic way conceived to correspond to the human souL 
Philolaus believed that 5 is the cause of color, 6 of cold, 7 of 
mind and health and light, 8 of love and friendship.^ In 
Plato's works are evidences of a similar belief in religiouB 
relations of numbers. Even Aristotle referred the virtues to 
numbers. 

Enough has been said about these mystic speculations to 
show what lively interest in mathematics they must have 
created and maintained. Avenues of mathematical inquiry 
were opened up by them which otherwise would probably 
have remained closed at that time. 

The Pythagoreans classified numbers into odd and even. 
They observed that the sum of the series of odd numbers 
from 1 to 2 n + 1 was always a complete square, and that 
by addition of the even numbers arises the series 2, 6, 12, 20, 
in which every number can be decomposed into two factors 
differing from each other by unity. Thus, 6 = 2»3, 12 = 3*4, 
etc. These latter numbers were considered of sufficient im- 
portance to receive the separate name of heteromecic (not 

n(n -4- 1^ 
equilateral).^ Numbers of the form ^ 7^ — ^ were called tri- 

angulaTy because they could always be arranged thus, ff^ . 

Numbers which were equal to the sum of all their possible 
factors, such as 6, 28, 496, were called perfect; those exceed- 
ing that sum, excessive; and those which were less, defective. 
Amicable numbers were those of which each was the sum of 
the factors in the other. Much attention was paid by the 
Pythagoreans to the subject of proportion. The quantities 
a, 6, c, d were said to be in arithmetical proportion when 
a — b = c — d; in geometrical proportion, when a :b = c:d'j in 
haj^monic proportion, when a -- b :b ~- c = a: c. It is probable 
that the Pythagoreans were also familiar with the mtLsicdl 



THB GREEKS. 69 

proportion a : ^ ^ = : 6. lamblichus says that Py- 
thagoras introduced it from Babylon. 

In connection with arithmetic, Pythagoras made extensive 
inTestigations into geometry. He believed that an arithmet- 
ical fact bad its analogue in geometry, and vice versa. In 
connection with his theorem on the right triangle he de- 
riaed a role by which integral numbers could be found, such 
tibat the sum of the squares of two of them equalled the 
■qnare of the third. Thus, take for one side an odd number 

(2n -♦- 1) ; then ^^^"^^)'"" -^ = 271' -f 2n = the other side, and 

(2ii^ -I- 2 n 4- 1) = hypotenuse. If 2 n -f- 1 = 9, then the other 

two numbers are 40 and 41. But this rule only applies to 

eases in which the hypotenuse differs from one of the sides 

by 1 In the study of the right triangle there doubtless arose 

qnestions of puzzling subtlety. Thus, given a number equal 

to the side of an isosceles right triangle, to find the number 

which the hypotenuse is equal to. The side may have been 

taken equal to 1, 2, f, ^, or any other number, yet in every 

instance all efforts to find a number exactly equal to the 

hypotenuse must have remained fruitless. The problem may 

^▼c been attacked again and again, until finally '^ some rare 

S^^ to whom it is granted, during some happy moments, 

t*> Mar with eagle's flight above the level of human think- 

^^%ii grasped the happy thought that this problem cannot be 

•olred. In some such manner probably arose the theory of 

'""WionoZ quantities, which is attributed by Eudemus to the 

Pythagoreans. It was indeed a thought of extraordinary 

^'^^ess, to assume that straight lines could exist, differing 

™^ one another not only in length, — that is, in quantity, — 

"^t also in a quality, which, though real, was absolutely 

^rtsible.' Need we wonder that the Pythagoreans saw in 



70 A HISTORY OF MATHJdCATICS. 

irrationals a deep mystery, a symbol of the unspeakable? 
We are told that the one who first divulged the theory of 
irrationals, which the Pythagoreans kept secret, perished in 
consequence in a shipwreck. Its discovery is ascribed to 
Pythagoras, but we must remember that all important Py- 
thagorean discoveries were, according to Pythagorean custom, 
referred back to him. The first incommensurable ratio known 
seems to have been that of the side of a square to its diagonal, 
as 1 : V2. Theodoras of Cyrene added to this the fact that the 
sides of squares represented in length by V3, V5, etc., up to 
Vl7, and Thesetetus, that the sides of any square, represented 
by a surd, are incommensurable with the linear unit. Euclid 
(about 300 B.C.), in his Elements, X. 9, generalised still further : 
Two magnitudes whose squares are (or are not) to one another 
as a square number to a square number are commensurable 
(or incommensurable), and conversely. In the tenth book, he 
treats of incommensurable quantities at length. He investi- 
gates every possible variety of lines which can be represented 
by -y/Va± Vb, a and b representing two commensurable lines, 
and obtains 25 species. Every individual of every species 
is incommensurable with all the individuals of every other 
species. " This book," says De Morgan, " has a completeness 
which none of the others (not even the fifth) can boast of; 
and we could almost suspect that Euclid, having arranged his 
materials in his own mind, and having completely elaborated 
the tenth book, wrot^ the preceding books after it, and did not 
live to revise them thoroughly." • The theory of incommen- 
surables remained where Euclid left it, till the fifteenth 
century. 

Euclid devotes the seventh, eighth, and ninth books of his 
Elements to arithmetic. Exactly how much contained in 
these books is Euclid's own invention, and how much is 
borrowed from his predecessors, we have no means of knowing. 



THE GREEKS. 71 

Without doubti much is original with Euclid. The seventh 
book begins with twenty-one definitions. All except that for 
'prime' numbers are known to have been given by the 
Pyths^reans. Next follows a process for finding the 
6.C.D. of two or more numbers. The eighth book deals with 
numbers in continued proportion, and with the mutual rela- 
tions of squares, cubes, and plane numbers. Thus, XXII., if 
three numbers are in continued proportion, and the first is a 
square, so is the third. In the ninth book, the same subject is 
eontmued. It contains the proposition that the number of 
primes is greater than any given number. 

After the death of Euclid, the theory of numbers remained 
almost stationary for 400 years. Geometry monopolised 
the attention of all Greek mathematicians. Only two are 
known to have done work in arithmetic worthy of mention. 
Intotthe&es (275-194 b.c.) invented a ' sieve ^ for finding prime 
numbers. All composite numbers are 'sifted^ out in the 
following manner : Write down the odd numbers from 3 up, 
in succession. By striking out every third number after the 
S, wa remove all multiples of 3. By striking out every fifth 
number after the 5, we remove all multiples of 5. In this way, 
by rejecting multiples of 7, 11, 13, etc., we have left prime 
numbers only. Hypsides (between 200 and 100 b.c.) worked 
at the subjects of polygonal numbers and arithmetical pro- 
gressions, which Euclid entirely neglected, in his work on 
'risings of the stars,' he showed (1) that in an arithmetical 
series of 2 n terms, the sum of the last n terms exceeds the sum 
of the first n by a multiple of n' ; (2) that in such a series of 
2n + 1 terms, the sum of the series is the number of terms 
multiplied by the middle term ; (3) that in such a series of 
2n terms, the sum is half the number of terms multiplied by 
the two middle terms.* 

For two centuries after the time of Hypsides, arithmetic 



72 A HISTORY OF MATHEMATICS. 

disappears from history. It is brought to light again about 
100 A.D. by Nicomachus, a Neo-Pythagorean, who inaugurated 
the final era of Greek mathematics. From now on, arithmetic 
was a favourite study, while geometry was neglected. Nico- 
machus wrote a work entitled Introductio Arithmetical which 
was very famous in its day.. The great number of commen- 
tators it has received vouch for its popularity. Boethius 
translated it into Latin. Lucian could pay no higher 
compliment to a calculator than this: ''You reckon like 
Nicomachus of Gerasa." The Introductio ArUhmetica was 
the first exhaustive work in which arithmetic was treated 
quite independently of geometry. Instead of drawing lines, 
like Euclid, he illustrates things by real numbers. To be 
sure, in his book the old geometrical nomenclature is retained, 
but the method is inductive instead of deductive. '' Its sole 
business is classification, and all its classes are derived from, 
and exhibited by, actual numbers." The work contains few 
results that are really original. We mention one important 
proposition which is probably the author^s own. He states 
that cubical numbers are always equal to the sum of successive 
odd numbers. Thus, 8 = 2« = 3 + 5, 27 = 3' = 7-f 9 -f H, 64 
= 4' = 13 -f 15 -f 17 -f 19, and so on. This theorem was used 
later for finding the sum of the cubical numbers themselves. 
Theon of Smyrna is the author of a treatise on '' the mathe- 
matical rules necessary for the study of Plato." The work is 
ill arranged and of little merit. Of interest is the theorem, 
that every square number, or that number minus 1, is 
divisible by 3 or 4 or both. A remarkable discovery is a 
proposition given by lamblichus in his treatise on Pythagorean 
philosophy. It is founded on the observation that the 
Pythagoreans called 1, 10, 100, 1000, units of the first, second, 
third, fourth 'course' respectively. The theorem is this: 
If we add any three consecutive numbers, of which the highest 



TEIE GK££KS. 78 

is divisible by 3, then add the digits of that sum, then, again, 
the digits of that sum, and so on, the final sum will be 6. 
Thus, 614-62 4-63 = 186, 14-84-6=15, 14-5 = 6. This 
disoovery was the more remarkable, because the ordinary 
Greek numerical symbolism was much less likely to suggest 
any such property of numbers than our ''Arabic'^ notation 
would have been. 

The works of Nicomachus, Theon of Smyrna, Thymaridas, 
tnd others contain at times investigations of subjects which 
are really algebraic in their nature. Thymaridas in one place 
nses the Greek word meaning '' unknown quantity " in a way 
which would lead one to believe that algebra was not far 
distant. Of interest in tracing the invention of algebra are 
the arithmetical epigrams in the Palatine Anthology, which con- 
tain about fifty problems leading to linear equations. Before 
the introduction of algebra these problems were propounded 
as puzzles. A riddle attributed to Euclid and contained in 
the Anthology is to this effect : A mule and a donkey were 
walking along, laden with corn. The mule says to the 
donkey, "If you gave me one measure, I should carry twice 
as much as you. If I gave you one, we should both carry 
equal burdens. Tell me their burdens, most learned master 
of geometry." • 

It will be allowed, says Oow, that this problem, if authentic, 
was not beyond Euclid, and the appeal to geometry smacks of 
antiquity. A far more difficult puzzle was the famous ' cattle- 
problem,' which Archimedes propounded to the Alexandrian 
mathematicians. The problem is indeterminate, for from only 
seven equations, eight unknown quantities in integral numbers 
are to be found. It may be stated thus : The sun had a herd 
of bulls and cows, of different colors. (1) Of Bulls, the 
white ( W) were, in number, {\ 4- \) of the blue (JB) and yel- 
low ( F) : the £ were {\ 4- i) of the Y and piebald (P) : the 



74 A HISTOKY OF MATHEMATICS. 

P were (\ -f \) of the TTand Y. (2) Of Cows, which had the 
same colors (to, 6, y, p), 

«^=a-Hi)(^-f6):6=(i + i)(P+l>):i) = a + i)(r-hy) 

•y=(i-Hl)(w^+«^). 

Find the number of bulls and cows/ Another problem in the 
ArUhology is quite familiar to school-boys: ''Of four pipes, 
one fills the cistern in one day, the next in two days, the third 
in three days, the fourth in four days : if all run together, how 
soon will they fill the cistern?" A great many of these 
problems, puzzling to an arithmetician, would have been solved 
easily by an algebraist. They became very popular about the 
time of Diophantus, and doubtless acted as a powerful stimu- 
lus on his mind. 

Diophantus was one of the last and most fertile mathema- 
ticians of the second Alexandrian school. He died about 
330 A.D. His age was eighty-four, as is known from an epitaph 
to this effect : Diophantus passed \ of his life in childhood, ^ in 
youth, and \ more as a bachelor; five years after his marriage 
was born a son who died four years before his father, at half 
his father's age. The place of nativity and parentage of 
Diophantus are unknown. If his works were not written in 
Greek, no one would think for a moment that they were the 
product of Greek mind. There is nothing in his works that 
reminds us of the classic period of Greek mathematics. His 
were almost entirely new ideas on a new subject. In the 
circle of Greek mathematicians he stands alone in his spe- 
cialty. Except for him, we should be constrained to say that 
among the Greeks algebra was always an unknown science. 

Of his works we have lost the PorismSy but possess a frag- 
ment of Polygonal Numbers, and seven books of his great work 
on Arithmetical said to have been written in 13 books. 

If we except the Ahmes papyrus, which contains the first 



THE GREEKS. 75 

suggestions of algebraic notation, and of the solution of 
equations, then his Arithmetica is the earliest treatise on 
algebra now extant. In this work is introduced the idea of 
an algebraic equation expressed in algebraic symbols. His 
treatment is purely analytical and completely divorced from 
geometrical methods. He is, as far as we know, the first to 
itate that ^'a negative number multiplied by a negative num- 
ber gives a positive number." This is applied to the multi- 
plication of differences, such as (a; — 1) (a; — 2). It must be 
lemarked, however, that Diophantus had no notion whatever 
of negative numbers standing by themselves. All he knew 
were differences, such as (2 a; — 10), in which 2 x could not be 
smaller than 10 without leading to an absurdity. He appears 
to be the first who could perform such operations as (x — 1) 
x(x — 2) without reference to geometiy. Such identities as 
(a -f 6)* = a* -f 2 a6 -f 6*, which with Euclid appear in the ele- 
vated rank of geometric theorems, are with Diophantus the 
simplest consequences of the algebraic laws of operation. His 
•ign for subtraction was >A, for equality i, For unknown 
quantities he had only one symbol, s. He had no sign for 
addition except juxtaposition. Diophantus used but few sym- 
bols, and sometimes ignored even these by describing an oper- 
ation in words when the symbol would have answered just 
as well. 

In the solution of simultaneous equations Diophantus adroitly 
managed with only one symbol for the unknown quantities and 
arrived at answers, most commonly, by the method of tentative 
assumption, which consists in assigning to some of the unknown 
quantities preliminary values, that satisfy only one or two of 
the conditions. These values lead to expressions palpably 
wrong, but which generally suggest some stratagem by which 
values can be secured satisfying all the conditions of the 
problem. 



76 A HISTORY OF MATHEMATICS. 

Diophantus also solved determinate equations of the second 
degree. We are ignorant of his method, for he nowhere goes 
through with the whole process of solution, but merely states 
the result. Thus, " 84 jb* -f 7 x = 7, whence x is found = J." 
Notice he gives only one root. His failure to observe that a 
quadratic equation has two roots, even when both roots are 
positive, rather surprises us. It must be remembered, how- 
ever, that this same inability to perceive more than one out of 
the several solutions to which a problem may point is common 
to all Greek mathematicians. Another point -to be observed 
is that he never accepts as an answer a quantity which is 
negative or irrational. 

Diophantus devotes only the first book of his Arithmetica to 
the solution of determinate equations. The remaining books 
extant treat mainly of indeterminate quadratic equations of the 
form Aa:^'\'Bx-{'C=y^y or of two simultaneous equations of the 
same form. He considers several but not all the possible 
cases which may arise in these equations. The opinion of 
Nesselmann on the method of Diophantus, as stated by Gow, 
is as follows : " (1) Indeterminate equations of the second 
degree are treated completely only when the quadratic or 
the absolute term is wanting: his solution of the equations 
Aa:^'{' C^y* and -4aj*-f JBa?-f C=i^ is in many respects cramped. 
(2) For the * double equation' of the second degree he has a 
definite rule only when the quadratic term is wanting in both 
expressions : even then his solution is not general. More com- 
plicated expressions occur only under specially favourable 
circumstances." Thus, he solves Bx-\- C^=z y*, BiX 4- Ci = yi. 

The extraordinary ability of Diophantus lies rather in 
another direction, namely, in his wonderful ingenuity to re- 
duce all sorts of equations to particular forms which he knows 
how to solve. Very great is the variety of problems considered. 
The 130 problems found in the great work of Diophantus con- 



THE BOMANS. 77 

over 50 different classes of problems, which are stning 
together without any attempt at classification. But still more 
multifarious than the problems are the solutions. General 
methods are unknown to Diophantus. Each problem has its 
own distinct method, which is often useless for the most 
doeely related problems. "It is, therefore, difficult for a 
modem, after studying 100 Diophantine solutions, to solve 
the 101st."' 

That M^hich robs his work of much of its scientific value is 
the fact that he always feels satisfied with one solution, though 
his equation may admit of an indefinite number of values. 
Another great defect is the absence of general methods. Mod- 
em mathematicians, such as Euler, La Grange, Gauss, had to 
begin the study of indeterminate analysis anew and received 
no direct aid from Diophantus in the formulation of methods. 
In spite of these defects we cannot fail to admire the work 
for the wonderful ingenuity exhibited therein in the solution 
of particular equations. 

It is still an open question and one of great difficulty 
whether Diophantus derived portions of his algebra from 
Hindoo sources or not. 



THE ROMANS. 

Nowhere is the contrast between the Greek and Roman 
mind shown forth more distinctly than in their attitude toward 
the mathematical science. The sway of the Greek was a 
flowering time for mathematics, but that of the Roman a 
period of sterility. In philosophy, poetry, and art the Roman 
was an imitator. But in mathematics he did not even rise to 
the desire for imitation. The mathematical fruits of Greek 
genius lay before him untasted. In him a science which had 



78 A HISTOBY OF MATHBlfATICS. 

no direct bearing on practical life could awake no interest. 
As a consequence, not only the higher geometry of Archimedes 
and Apollonius, but even the Elements of Euclid, were en- 
tirely neglected. What little mathematics the Romans pos- 
sessed did not come from the Greeks, but from more ancient 
^ sources. Exactly where and how it originated is a matter of 
doubt. It seems most probable that the '^ Eoman notation/' 
as well as the practical geometry of the Eomans, came from 
the old Etruscans, who, at the earliest period to which our 
knowledge of them extends, inhabited the district between the 
Amo and Tiber. 

Livy tells us that the Etruscans were in the habit of repre- 
senting the number of years elapsed, by driving yearly a nail 
into the sanctuary of Minerva, and that the Eomans continued 
this practice. A less primitive mode of designating numbers, 
presumably of Etruscan origin, was a notation resembling the 
present " Eoman notation." This system is noteworthy from 
the fact that a principle is involved in it which is not met 
with in any other ; namely, the principle of subtraction. If a 
letter be placed before another of greater value, its value is 
not to be added to, but subtracted from, that of the greater. 
In the designation of large numbers a horizontal bar placed 
over a letter was made to increase its value one thousand fold. 
In fractions the Bomans used the duodecimal system. 

Of arithmetical calculations, the Eomans employed three 
different kinds : Reckoning on the fingers, upon the abacus, 
and by tables prepared for the purpose.' Finger-symbolism 
was known as early as the time of King Numa, for he had 
erected, says Pliny, a statue of the double-faced Janus, of 
which the fingers indicated 365 (355?), the number of days in 
a year. Many other passages from Roman authors point out 
the use of the fingers as aids to calculation. In fact, a finger- 
symbolism of practically the same form was in use not only in 



THE ROMANS. 79 

Some, bat also in Greece and throughout the East, certainly 
as early as the beginning of the Christian era, and continued 
to be used in Earope during the Middle Ages. We possess no 
knowledge as to where or when it was invented. The second 
mode of calculation, by the abacus, was a subject of elemen- 
tary instruction in Rome. Passages in Koman writers indicate 
that the kind of abacus most commonly used was covered with 
dnst and then divided into columns by drawing straight lines. 
Each column was supplied with pebbles (calculi, whence * cal- 
eolaie' and ^calculate') which served for calculation. Addi- 
tions and subtractions could be performed on the abacus quite 
easily, but in multiplication the abacus could be used only for 
adding the particular products, and in division for performing 
the subtractions occurring in the process. Doubtless at this 
point recourse was made to mental operations and to the mul- 
tiplication table. Possibly finger-multiplication may also have 
been used. But the multiplication of large numbers must, by 
either method, have been beyond the power of the ordinary 
arithmetician. To obviate this difficulty, the arithmetical 
tables mentioned above were used, from which the desired 
prodncts could be copied at once. Tables of this kind were 
prepared by Victorius of Aquitania. His tables contain a 
peculiar notation for fractions, which continued in use through- 
out the Middle Ages. Victorius is best known for his canon 
paxhalia, a rule for finding the correct date for Easter, which 
he published in 457 a.d. 

Payments of interest and problems in interest were very old 
among the Eomans. The Eoman laws of inheritance gave 
rise to numerous arithmetical examples. Especially unique is 
the following : A dying man wills that, if his wife, being with 
child, gives birth to a son, the son shall receive | and she \ of 
his estates ; but if a daughter is born, she shall receive | and 
his wife f. It happens that twins are bom, a boy and a girl. 



80 A HISTORY OF MATHBMATICS. 

How shall the estates be divided so as to satisfy the will? 
The celebrated Roman jurist, Salvianus Julianas, decided that 
the estates shall be divided into seven equal parts, of which 
the son receives four, the wife two, the daughter one. 

We next consider Eoman geometry. He who expects to 
find in Eome a science of geometry, with definitions, axioms, 
theorems, and proofs arranged in logical order, will be disap- 
pointed. The only geometry known was 2i practical geometry, 
which, like the old Egyptian, consisted only of empirical rules. 
This practical geometry was employed in surveying. Treatises 
thereon have come down to us, compiled by the Eoman sur- 
veyors, called agrimensores or gromaticL One would naturally 
expect rules to be clearly formulated. But no ; they are left 
to be abstracted by the reader from a mass of numerical exam- 
ples. ''The total impression is as though the Eoman gromatic 
were thousands of years older than Greek geometry, and as 
though a deluge were lying between the two." Some of their 
rules were probably inherited from the Etruscans, but others 
are identical with those of Heron. Among the latter is that 
for finding the area of a triangle from its sides and the approx- 
imate formula, ^a% for the area of equilateral triangles (a 
being one of the sides). But the latter area was also calculated 
by the formulas ^^^a^-^-a) and ^a', the first of which was 
unknown to Heron. Probably the expression ^* was derived 

from the Egyptian formula ^^ — "^ for the determina- 

tion of the surface of a quadrilateral. This Egyptian formula 
was used by the Eomans for finding the area, not only of rec- 
tangles, but of any quadrilaterals whatever. Indeed, the groma- 
tici considered it even sufficiently accurate to determine the 
areas of cities, laid out irregularly, simply by mieasuring their 
circumferences.^ Whatever Egyptian geometry the Eomans 
possessed was transplanted across the Mediterranean at the 



THE B0MAN8. 81 

time of Julius CcBsar, who ordered a survey of the whole 
empire to secure an equitable mode of taxation. Caesar also 
reformed the calendar^ and, for that purpose, drew from 
Egyptian learning. He secured the services of the Alexan- 
drian astronomer, Sosigenea. 

In the fifth century, the Western Roman Empire was fast 
Mling to pieces. Three great branches — Spain, Gaul, and 
the province of Africa — broke off from the decaying trunk. 
In 476, the Western Empire passed away, and the Yisigothic 
chief, Odoacer, became king. Soon after, Italy was conquered 
by the Ostrogoths under Theodoric. It is remarkable that 
this very period of political humiliation should be the one 
duriDg which Greek science was studied in Italy most zeal- 
ously. School-books began to be compiled from the elements 
of Greek authors. These compilations are very deficient, but 
are of absorbing interest, from the fact that, down to the 
twelfth century, they were the only sources of mathematical 
knowledge in the Occident. Foremost among these writers is 
Boethint (died 524). At first he was a great favourite of King 
Tkeodoric, but later, being charged by envious courtiers with 
^^eason, he was imprisoned, and at last decapitated. While 

• 

^ prison he wrote On the Consolations of Philosophy. As a 
mathematician, Boethius was a Brobdingnagian among Roman 
^^lars, but a Liliputian by the side of Greek masters. He 
^^'ote an Jjistitutis Arithmetica, which is essentially a transla- 
tion of the arithmetic of Nicomachus, and a Geometry in 
*^^eral books. Some of the most beautiful results of Nico- 
Diachus are omitted in Boethius' arithmetic. The first book 
<>n geometry is an extract from Euclid^s Elements, which con- 
^08, in addition to definitions, postulates, and axioms, the 
theorems in the first three books, without proofs. How can 
this omission of proofs be accounted for ? It has been argued 
hy some that Boethius possessed an incomplete Greek copy of 



82 A HISTORY OF MATHEliATICS. 

the Elements; by others, that he had Theon's edition before 
him, and believed that only the theorems came from Euclid, 
while the proofs were supplied by Theon. The second book, 
as also other books on geometry attributed to Boethius, 
teaches, from numerical examples, the mensuration of plane 
figures after the fashion of the agrimensores. 

A celebrated portion in the geometry of Boethius is that 
pertaining to an abacus, which he attributes to the Pythago- 
reans. A considerable improvement on the old abacus is 
there introduced. Pebbles are discarded, and apices (probably 
small cones) are used. Upon each of these apices is drawn 
a numeral giving it some value below 10. The names of 
these numerals are pure Arabic, or nearly so, but are added, 
apparently, by a later hand. These figures are obviously the 
parents of our modern "Arabic" numerals. The is not 
mentioned by Boethius in the text. These numerals bear 
striking resemblance to the Gubar-numerals of the West- 
Arabs, which are admittedly of Indian origin. These facts 
have given rise to an endless controversy. Some contended 
that Pythagoras was in India, and from there brought the 
nine numerals to Greece, where the Pythagoreans used them 
secretly. This hypothesis has been generally abandoned, for 
it is not certain that Pythagoras or any disciple of his ever 
was in India, nor is there any evidence in any Greek author, 
that the apices were known to the Greeks, or that numeral 
signs of any sort were used by them with the abacus. It is 
improbable, moreover, that the Indian signs, from which the 
apices are derived, are so old as the time of Pythagoras. 
A second theory is that the Oeometry attributed to Boethius 
is a forgery ; that it is not older than the tenth, or possibly 
the ninth, century, and that the apices are derived from the 
Arabs. This theory is based on contradictions between pas- 
sages in the ArithmeHca and others in the Oeometry. But 



THE ROMANS. 88 

there is an Encjclopsedia written by Cassiodorius (died about 
570) in which both the arithmetic and geometry of Boethius 
are mentioned. There appears to be no good reason for doubt- 
ing the trustworthiness of this passage in the Encyclopaedia. 
A third theory (Woepcke's) is that the Alexandrians either 
directly or indirectly obtained the nine numerals from the 
Hindoos, about the second century a.d., and gave them to 
the Romans on the one hand, and to the Western Arabs 
on the other. This explanation is the most plausible. 



MIDDLE AGES. 



THE HINDOOS. 

The first people who distinguished themselves in mathe- 
matical research, after the time of the ancient Greeks, belongedi 
like them, to the Aryan race. It was, however, not a Euro- 
pean, but an Asiatic nation, and had its seat in far-off India. 

Unlike the Greek, Indian society was fixed into castes. The 
only castes enjoying the privilege and leisure for advanced 
study and thinking were the Brahmins^ whose prime business 
was religion and philosophy, and the Kshatriyas, who attended 
to war and government. 

Of the development of Hindoo mathematics we know but 
little. A few manuscripts bear testimony that the Indians 
had climbed to a lofty height, but their path of ascent is no 
longer traceable. It would seem that Greek mathematics grew 
up under more favourable conditions than the Hindoo, for in 
Greece it attained an independent existence, and was studied 
for its own sake, while Hindoo mathematics always remained 
merely a servant to astronomy. Furthermore, in Greece 
mathematics was a science of the people, free to be cultivated 
by all who had a liking for it ; in India, as in Egypt, it was in 
the hands chiefly of the priests. Again, the Indians were in 
the habit of putting into verse all mathematical results they 
obtained, and of clothing them in obscure and mystic language, 

84 



THE HINDOOS. 85 

whiehy though well adapted to aid the memory of him who 

already miderstood the subject, was ofteu unintelligible to the 

minitiated. Although the great Hindoo mathematicians 

doabUess reasoned out most or all of their discoveries, yet 

tiiey were not in the habit of preserving the proofs, so that 

tiie naked theorems and processes of operation are all that 

have oome down to our time. Very different in these respects 

were the Greeks. Obscurity of language was generally 

aToided, and proofs belonged to the stock of knowledge quite 

as much as the theorems themselves. Very striking was the 

difference in the bent of mind of the Hindoo and Greek ; for, 

while the Greek mind was pre-eminently geometrical^ the 

Indian was first of all arithmeticaX, The Hindoo dealt with 

number, the Greek with form. Numerical symbolism, the 

science of numbers, and algebra attained in India far greater 

perfection than they had previously reached in Greece. On 

the other hand, we believe that there was little or no geom- 

etiy in India of which the source may not be traced back to 

Greece. Hindoo trigonometry might possibly be mentioned 

as an exception, but it rested on arithmetic more than on 

geometry. 

An interesting but difficult task is the tracing of the rela- 
tion between Hindoo and Greek mathematics. It is well 
known that more or less trade was carried on between Greece 
and India from early times. After Egypt had become a 
Roman province, a more lively commercial intercourse sprang 
up between Rome and India, by way of Alexandria. A priori^ 
it does not seem improbable, that with the traffic of merchan- 
dise there should also be an interchange of ideas. That 
communications of thought from the Hindoos to the Alexan- 
drians actually did take place, is evident from the fact that 
certain philosophic and theologic teachings of the Manicheans, 
Neo-Platonists, Gnostics, show immistakable likeness to 



86 A HISTORY OF MATHEMATICS. 

Indian tenets. Scientific facts passed also from Alexandria 
to India. This is shown plainly by the Greek origin of some 
of the technical terms used by the Hindoos. Hindoo astron- 
omy was influenced by Greek astronomy. Most of the geo- 
metrical knowledge which they possessed is traceable to 
Alexandria, and to the writings of Heron in particular. In 
algebra there was, probably, a mutual giving and receiving. 
We suspect that Diophantus got the first glimpses of algebraic 
knowledge from India. On the other hand, evidences have 
been found of Greek algebra among the Brahmins. The 
earliest knowledge of algebra in India may possibly have been 
of Babylonian origin. When we consider that Hindoo scien- 
tists looked upon arithmetic and algebra merely as tools 
useful in astronomical research, there appears deep irony in 
the fact that these secondary branches were after all the only 
ones in which they won real distinction, while in their pet 
science of astronomy they displayed an inaptitude to observe, 
to collect facts, and to make inductive investigations. 

We shall now proceed to enumerate the names of the 
leading Hindoo mathematicians, and then to review briefly 
Indian mathematics. We shall consider the science only in 
its complete state, for our data are not sufficient to trace the 
history of the development of methods. Of the great Indian 
mathematicians, or rather, astronomers, — for India had no 
mathematicians proper, — Aryabhatta is the earliest. He was 
born 476 a.d., at Pataliputra, on the upper Ganges. His 
celebrity rests on a work entitled Aryabhattiyamy of which 
the third chapter is devoted to mathematics. About one 
hundred years later, mathematics in India reached the highest 
mark. At that time flourished Brahmagupta (born 598). In 
628 he wrote his Brahmorsphutarsiddhanta ("The Revised Sys- 
tem of Brahma"), of which the twelfth and eighteenth chapters 
belong to mathematics. To the fourth or fifth century belongs 



THE HINDOOS. 87 

anonymous astronomical work, called Suryorsiddhanta 
C Knowledge from the Sim"), which by native authorities 
was ranked second only to the Brahma-siddhantaf but is of in- 
terest to ns merely as furnishing evidence that Greek science 
influenced Indian science even before the time of Aryabhatta. 
The following centuries produced only two names of impor- 
tance; namely, Cridhara, who wrote a Oanitorsara ('' Quintes- 
sence of Calculation"), and Padmanabha, the author of an 
i^bra. The science seems to have made but little progress 
it this time ; for a work entitled Siddhantaciromani {" Diadem 
of an Astronomical System "), written by Bhaskara Acarya in 
1150, stands little higher than that of Brahmagupta, written 
over 500 years earlier. The two most important mathematical 
chapters in this work are the LUavati (= ''the beautiful," i.e. 
the noble science) and Viga-ganita (= " root-extraction "), de- 
moted to arithmetic and algebra. From now on, the Hindoos 
in the Brahmin schools seemed to content themselves with 
rtadying the masterpieces of their predecessors. Scientific 
hitelligence decreases continually, and in modem times a very 
^ficient Arabic work of the sixteenth century has been held 

• 

*n great authority.'' 

lie mathematical chapters of the Brahmorsiddhanta and 
^^^dhantoLciromani were translated into English by H. T. 
Colebrooke, London, 1817. The Surya-siddhanta was trans- 
^^^ by E. Burgess, and annotated by W. D. Whitney, New 
Haven, Conn., 1860. 

The grandest achievement of the Hindoos and the one 
^hich, of all mathematical inventions, has contributed most 
^ the general progress of intelligence, is the invention of 
^* principle of position in writing numbers. Generally we 
"peak of our notation as the " Arabic " notation, but it should 
^ called the " Hindoo " notation, for the Arabs borrowed it 
^'om the Hindoos. That the invention of this notation was 



88 A HISTORY OF MATHEMATICS. 

not so easy as we might suppose at first thought^ may be 
inferred from the fact that, of other nations, not even the 
keen-minded Greeks possessed one like it. We inquire, who 
invented this ideal symbolism, and when? But we know 
neither the inventor nor the time of invention. That our 
system of notation is of Indian origin is the only point of 
which we are certain. From the evolution of ideas in general 
we may safely infer that our notation did not spring into 
existence a completely armed Minerva from the head of 
Jupiter. The nine figures for writing the units are supposed 
to have been introduced earliest, and the sign of zero and the 
principle of position to be of later origin. This view receives 
support from the fact that on the island of Ceylon a notation 
resembling the Hindoo, but without the zero has been pre- 
served. We know that Buddhism and Indian culture were 
transplanted to Ceylon about the third century after Christ, 
and that this culture remained stationary there, while it made 
progress on the continent. It seems highly probable, then, 
that the numerals of Ceylon are the old, imperfect numerals 
of India. In Ceylon, nine figures were used for the units, 
nine others for the tens, one for 100, and also one for 1000. 
These 20 characters enabled them to write all the numbers up 
to 9999. Thus, 8725 would have been written with six signs, 
representing the following numbers : 8, 1000, 7, 100, 20, 5. 
These Singhalesian signs, like the old Hindoo numerals, are 
supposed originally to have been the initial letters of the corre- 
sponding numeral adjectives. There is a marked resemblance 
between the notation of Ceylon and the one used by Aryabhatta 
in the first chapter of his work, and there only. Although the 
zero and the principle of position were unknown to the scholars 
of Ceylon, they were probably known to Aryabhatta; for, in 
the second chapter, he gives directions for extracting the square 
and cube roots, which seem to indicate a knowledge of them. 



THE HINDOOS. 89 

It would appear that the zero and the accompanying pnnciple 
of position were introduced about the time of Aryabhatta. 
These are the inventions which give the Hindoo system its 
great superiority, its admirable perfection. 

There appear to have been several notations in use in 

different parts of India, which differed, not in principle, but 

merely in the forms of the signs employed. Of interest is 

also a symbolical system of position^ in which the figures 

generally were not expressed by numerical adjectives, but by 

objects suggesting the particular numbers in question. Thus, 

for 1 were used the words m^oouy Brahma, Creator, or form; 

for 4, the words Veda, (because it is divided into four parts) 

or ocean, etc. The following example, taken from the Surya- 

•iddhanta, illustrates the idea. The number 1,577,917,828 is 

pressed from right to left as follows: Yasu (a class of 8 

S^) -h two 4- eight + mountains (the 7 mountain-chains) 

+ form 4- digits (the 9 digits) + seven -f mountains 4- lunar 

^78 (half of which equal 15). The use of such notations 

'^^e it possible to represent a number in several different 

^ys. This greatly facilitated the framing of verses con- 

^tiing arithmetical rules or scientific constants, which could 

^Xis be more easily remembered. 

^t an early period the Hindoos exhibited great skill in 

^^Iculating, even with large numbers. Thus, they tell us of 

^^ examination to which Buddha, the reformer of the Indian 

^ligion, had to submit, when a youth, in order to win the 

^^aiden he loved. In arithmetic, after having astonished his 

Examiners by naming all the periods of numbers up to the 

'^dd, he was asked whether he could determine the number 

^f primary atoms which, when placed one against the other, 

Vould form a line one mile in length. Buddha found the 

required answer in this way : 7 primary atoms make a very 

minnte grain of dust, 7 of these make a minute grain of dust. 



90 A HISTOBY OF MATHEMATICS. 

7 of tJiese a grain of dust whirled up by the wind, and so on. 
Thus he proceeded, step by step, until he finally reached the 
length of a mile. The multiplication of all the factors gave 
for the multitude of primary atoms in a mile a number con- 
sisting of 15 digits. This problem reminds one of the < Sand- 
Counter ' of Archimedes. 

After the numerical symbolism had been perfected, figuring 
was made much easier. Many of the Indian modes of 
operation differ from ours. The Hindoos were generally 
inclined to follow the motion from left to right, as in writing. 
Thus, they added the left-hand columns first, and made the 
necessary corrections as they proceeded. For instance, they 
would have added 254 and 663 thus : 2 -f- 6 = 8, 5 -f- 6 = 11, 
which changes 8 into 9, 4 -f- 3 = 7. Hence the sum 917. In 
subtraction they had two methods. Thus in 821 — 348 they 
would say, 8 from 11 = 3, 4 from 11 = 7, 3 from 7 = 4. Or 
they would say, 8 from 11 = 3, 5 from 12 = 7, 4 from 8 = 4. 
In multiplication of a number by another of only one digit, say 
569 by 5, they generally said, 5«5 = 25, 5-6 = 30, which 
changes 25 into 28, 5*9 = 45, hence the must be increased by 
4. The product is 2845. In the multiplication with each 
other of many-figured numbers, they first multiplied, in the 
manner just indicated, with the left-hand digit of the multi- 
plier, which was written above the multiplicand, and placed 
the product above the multiplier. On multiplying with the 
next digit of the multiplier, the product was not placed in 
a new row, as with us, but the first product obtained was 
corrected, as the process continued, by erasing, whenever 
necessary, the old digits, and replacing them by new ones, 
until finally the whole product was obtained. We who possess 
the modern luxuries of pencil and paper, would not be likely 
to fall in love with this Hindoo method. But the Indians 
wrote " with a cane-pen upon a small blackboard with a white, 



THB HINDOOS. 



91 



/ 


/'. 


/\ 


X 


/ 


X 



8 



8 



2 



thinly liquid paint which made marks that could be easily 
erased, or upon a white tablet, less than a foot square, strewn 
with red flour, on which they wrote the figures with a small 
stick, so that the figures appeared white on a red ground." ^ 
Since the digits had to be quite large to be distinctly legible, 
and since the boards were small, it was desirable to have a 
method which would not require much space. Such a one 
was the above method of multiplication. Figures could be 
easOy erased and replaced by others without sacrificing neat- 
ness. But the Hindoos had also other 

7 3ft 

ways of multiplying, of which we men- 
tion the following: The tablet was 
divided into squares like a chess-board. 
Diagonals were also drawn, as seen in 
the figure. The multiplication of 12 x 
7% = 8820 is exhibited in the adjoining diagram.' The man- 
^ripts extant give no information of how divisions were 
®*^uted. The correctness of their additions, subtractions, 
*^d multiplications was tested " by excess of 9's." In writing 
^^tions, the numerator was placed above the denominator, 
^t no line was drawn between them. 

We shall now proceed to the consideration of some arith- 
metical problems and the Indian modes of solution. A favour- 
^^ method was that of inversion. With laconic brevity, 
'^ryabhatta describes it thus : " Multiplication becomes divis- 
^t>n, division becomes multiplication ; what was gain becomes 
loss, what loss, gain; inversion.'^ Quite different from this 
tjnotation in style is the following problem from Aryabhatta, 
which illustrates the method : ' *^ Beautiful maiden with beam- 
ing eyes, tell me, as thou understandst the right method of 
inversion, which is the number which multiplied by 3, then 
increased by J of the product, divided by 7, diminished by \ of 
the quotient, multiplied by itself, diminished by 52, the square 



92 A HISTOBY OF MATHEMATICS. 

root extracted, addition of 8, and division by 10, gives the 
number 2?" The process consists in beginning with 2 and 
working backwards. Thus, (2 • 10-8)*+52=196, Vi96=14, 
and 14 • f • 7 • ^ -f- 3 = 28, the answer. 

Here is another example taken from LUavaH, a chapter in 
Bhaskara's great work : " The square root of half the number 
of bees in a swarm has flown out upon a jessamine-bush, f of 
the whole swarm has remained behind ; one female bee flies 
about a male that is buzzing within a lotus-flower into which 
he was allured in the night by its sweet odour, but is now 
imprisoned in it. Tell me the number of bees." Answer, 72. 
The pleasing poetic garb in which all arithmetical problems 
are clothed is due to the Indian practice of writing all school- 
books in verse, and especially to the fact that these problems, 
propounded as puzzles, were a favourite social amusement. 
Says Brahmagupta : " These problems are proposed simply for 
pleasure ; the wise man can invent a thousand others, or he 
can solve the problems of others by the rules given here. As 
the sun eclipses the stars by his brilliancy, so the man of 
knowledge will eclipse the fame of others in assemblies of the 
people if he proposes algebraic problems, and still more if he 
solves them." 

The Hindoos solved problems in interest, discount, partner- 
ship, alligation, summation of arithmetical and geometric 
series, devised rules for determining the numbers of combina- 
tions and permutations, and invented magic squares. It may 
here be added that chess, the profoimdest of all games, had its 
origin in India. 

The Hindoos made frequent use of the " rule of three," and 
also of the method of " falsa positio," which is almost identi- 
cal with that of the "tentative assumption" of Diophantus. 
These and other rules were applied to a large number of 
problems. 



THE HINDOOS. 93 

Passing now to algebra, we shall first take up the symbols 
of operation. Addition was indicated simply by juxtaposition 
as in Diophantine algebra; subtraction, by placing a dot over 
the subtrahend ; multiplication, by putting after the factors, 
bhfif the abbreviation of the word bJmvita, " the product " ; 
division, by placing the divisor beneath the dividend ; square- 
root, by writing ka, from the word karana (irrational), before 
the quantity. The unknown quantity was called by Brahma- 
gapta ydvattdvat (qtiantum tantum) . When several unknown 
quantities occurred, he gave, imlike Diophantus, to each a 
distinct name and symbol The first unknown was designated 
by the general term "unknown quantity." The rest were 
distinguished by names of colours, as the black, blue, yellow, 
red, or green unknown. The initial syllable of each word 
constituted the symbol for the respective unknown quantity. 
Thus yd meant a;; kd (from kdlaka = hlsick) meant y; yd kd 
bha, " X tunes y'';ka 15 ka 10, « Vi5 - VlO." 

The Indians were the first to recognise the existence of 
absolutely negative quantities. They brought out the differ- 
ence between positive and negative quantities by attaching to 
the one the idea of < possession,' to the other that of ' debts.' 
The conception also of opposite directions on a line, as an 
interpretation of -f- and — quantities, was not foreign to them. 
They advanced beyond Diophantus in observing that a quad- 
ratic has always two roots. Thus Bhaskara gives a; = 50 and 
X = — 5 for the roots of «* — 45 a? = 250. " But," says he, 
^'the second value is in this case not to be taken, for it is 
inadequate ; people do not approve of negative roots." Com- 
mentators speak of this as if negative roots were seen, but not 
admitted. 

Another important generalisation, says Hankel, was this, 
that the Hindoos never confined their arithmetical operations 
to rational numbers. For instance, Bhaskara showed how. 



94 A HI8TOBY OF MATHEMATICS. 

by the fomiila V^T^ =-^^i^S +^J«Z^^ 

the square root of the sum of rational and irrational numbers 
could be found. The Hindoos never discerned the dividing 
line between numbers and magnitudes, set up by the Greeks, 
which, though the product of a scientific spirit, greatly re- 
tarded the progress of mathematics. They passed from mag- 
nitudes to numbers and from numbers to magnitudes without 
anticipating that gap which to a sharply discriminating mind 
exists between the continuous and discontinuous. Yet by 
doing so the Indians greatly aided the general progress of 
mathematics. "Indeed, if one undei*stands by algebra the 
application of arithmetical operations to con\plex magnitudes 
of all sorts, whether rational or irrational numbers or space- 
magnitudes, then the learned Brahmins of Hindostan are the 
real inventors of algebra." ' 

Let us now examine more closely the Indian algebra. In 
extracting the square and cube roots they used the formulas 
(a + 6) « = a« 4- 2 aft + y* and (a + by= a» -f 3 a«6 -f 3 a6« -f 6*. 
In this connection Aryabhatta speaks of dividing a number 
into periods of two and three digits. From this we infer that 
the principle of position and the zero in the numeral notation 
were already known to him. In figuring with zeros, a state- 
ment of Bhaskara is interesting. A fraction whose denomi- 
nator is zero, says he, admits of no alteration, though much be 
added or subtracted. Indeed, in the same way, no change 
takes place in the infinite and immutable Deity when worlds 
are destroyed or created, even though numerous orders of beings 
be taken up or brought forth. Though in this he apparently 
evinces clear mathematical notions, yet in other places he 
makes a complete failure in figuring with fractions of zero 
denominator. 

In the Hindoo solutions of determinate equations, Cantor 



THE HINDOOS. 95 

thinks he can see traces of Diophantine methods. Some 
technical terms betray their Greek origin. Even if it be true 
that the Indians borrowed from the Greeks, they deserve great 
credit for improving and generalising the solutions of linear 
and quadratic equations. Bhaskara advances far beyond the 
Greeks and even beyond Brahmagupta when he says that 
''the square of a positive, as also of a negative number, 
is positive; that the square root of a positive number is 
twofold, positive and negative. There is no square root 
of a negative number, for it is not a square.'^ Of equa- 
tions of higher degrees, the Indians succeeded in solving 
only some special cases in which both sides of the equation 
could be made perfect powers by the addition of certain 
terms to each. 

Incomparably greater progress than in the solution of deter- 
minate equations was made by the Hindoos in the treatment 
of indeterminate equations. Indeterminate analysis was a 
subject to which the Hindoo mind showed a happy adaptation. 
"We have seen that this very subject was a favourite with Dio- 
phantus, and that his ingenuity was almost inexhaustible in 
devising solutions for particular cases. But the glory of 
having invented general methods in this most subtle branch 
of mathematics belongs to the Indians. The Hindoo indeter- 
minate analysis differs from the Greek not only in method, 
but also in aim. The object of the former was to find all 
possible integral solutions. Greek analysis, on the other hand, 
demanded not necessarily integral, but simply rational answers. 
Diophantus was content with a single solution ; the Hindoos 
endeavoured to find all solutions possible. Aryabhatta gives 
solutions in integers to linear equations of the form ax± by=iC, 
where ayb,c are integers. The rule employed is called the 
pulveriser. For this, as for most other rules, the Indians give 
no proof. Their solution is essentially the same as the one of 



96 A HISTORY OF MATHEMATICS. 

Euler. Elder's process of reducing ~ to a continued fraction 



amounts to the same as the Hindoo process of finding the 
greatest common divisor of a and b by division. This is fre- 
quently called the Diophantine method. Hankel protests 
against this name, on the ground that Diophantus not only 
never knew the method, but did not even aim at solutions 
purely integral.' These equations probably grew out of prob- 
lems in astronomy. They were applied, for instance, to 
determine the time when a certain constellation of the planets 
would occur in the heavens. 

Passing by the subject of linear equations with more than 
two unknown quantities, we come to indeterminate quadratic 
equations. In the solution of xy = ax + by -{- c, they applied 
the method re-invented later by Euler, of decomposing {ab + c) 
into the product of two integers m • n and of placing x = m + b 
and y = n 4- a. 

Brcmarkable is the Hindoo solution of the quadratic equa- 
tion cy'ss 005*4-6. With great keenness of intellect they 
recognised in the special case y^^sax^ + l a fundamental 
problem in indeterminate quadratics. They solved it by the 
cyclic method. " It consists,*' says De Morgan, " in a rule for 
finding an indefinite number of solutions of y* = aa^ 4- 1 (a be- 
ing an integer which is not a square), by means of one solution 
given or found, and of feeling for one solution by making a 
solution of y* = 005* -1-6 give a solution of y* = asc* -f- 6*. It 
amounts to the following theorem : If p and q be one set of 
values of x and y in y* = cux^ 4- b and p' and g' the same or 
another set, then qp + pq' and app* -f qq' are values of x and y 
in y* = aa^ -f 6*. From this it is obvious that one solution of 
y* =s oaj* -f- 1 may be made to give any number, and that if, 
taking b at pleasure, y* = aa? -f ^ can be solved so that x and y 
are divisible by 6, then one preliminary solution of y* = oos* -f- 1 



THE HINDOOS. 



97 



be found. Another mode of trying for solutions is a 
combination of the preceding with the cuttaca (pulveriser)." 
These calculations were used in astronomy. 

Doubtless this '^ cyclic method'' constitutes the greatest 
invention in the theory of numbers before the time of La- 
grange. The perversity of fate has willed it, that the equation 
y'ssaac'+l should now be called PeWs problem, while in 
recognition of Brahmin scholarship it ought to be called the 
^Hindoo problem/' It is a problem that has exercised the 
highest Acuities of some of our greatest modern analysts. 
By them the work of the Hindoos was done over again ; for, 
unfortunately, the Arabs transmitted to Europe only a small 
part of Indian algebra and the original Hindoo manuscripts, 
which we now possess, were unknown in the Occident. 

Hindoo geometry is far inferior to the Greek. In it are 
found no definitions, no postulates, no axioms, no logical 
chain of reasoning or rigid form of demonstration, as with 
Euclid. Each theorem stands by itself as an independent 
truth. Like the early Egyptian, it is empirical Thus, in the 
proof of the theorem of the right triangle, Bhaskara draws 
the right triangle four times in the square of the hypotenusCi 
80 that in the middle 
there remains a square 
whose side equals the 
difference between the 
two sides of the right 
triangle. Arranging this square and the four triangles in a 
different way, they are seen, together, to make up the sum 
of the square of the two sides. "Behold!" says Bhaskara, 
without adding another word of explanation. Bretscbneider 
conjectures that the Pythagorean proof was substantially the 
same as this. In another place, Bhaskara gives a second 
demonstration of this theorem by drawing from the vertex of 




I 



98 A HISTORY OF MATHEMATICS. 

the right angle a perpendicular to the hypotenuse^ and com- 
paring the two triangles thus obtained with the given triangle 
to which they are similar. This proof was unknown in Europe 
till Wallis rediscovered it. The Brahmins never inquired 
into the properties of figures. They considered only metrical 
relations applicable in practical life. In the Greek sense, the 
Brahmins never had a science of geometry. Of interest is 
the formula given by Brahmagupta for the area of a triangle 
in terms of its sides. In the great work attributed to Heron 
the Elder this formula is first found. Whether the Indians 
themselves invented it, or whetjjer they borrowed it from 
Heron, is a disputed question. Several theorems are given 
by Brahmagupta on quadrilaterals which are true only of 
those which can be inscribed on a circle — a limitation which 
he omits to state. Among these is the proposition of Ptole- 
maeus, that the product of the diagonals is equal to the sum 
of the products of the opposite sides. The Hindoos were 
familiar with the calculation of the areas of circles and their 
segments, of the length of chords and perimeters of regular 
inscribed polygons. An old Indian tradition makes ir = 3, 
also = ViO ; but Aryabhatta gives the value fHM- Bhaskara 
gives two values, — the 'accurate,* ff|^, and the 'inaccurate,' 
Archimedean value, ^. A commentator on lAlavati says 
that these values were calculated by beginning with a regu- 
lar inscribed hexagon, and applying repeatedly the formula 

AD = \2 — V 4 — Aff, wherein AB is the side of the given 
polygon, and AD that of one with double the number of 
sides. In this way were obtained the perimeters of the 
inscribed polygons of 12, 24, 48, 96, 192, 384 sides. Taking 
the radius = 100, the perimeter of the last one gives the value 
which Aryabhatta used for tt. 

Greater taste than for geometry was shown by the Hindoos 
for trigonometry. Like the Babylonians and Greeks, they 



THE HINDOOS. 99 

diyided the circle into quadrants, each quadrant into 90 de- 
grees and 5400 minutes. The whole circle was therefore 
made up of 21,600 equal parts. From Bhaskara's ^accurate' 
value for «• it was found that the radius contained 3438 of 
these circular parts. This last step was not Grecian. The 
Greeks might have had scruples about taking a part of a 
curve as the measure of a straight, line. Each quadrant was 
divided into 24 equal parts, so that each part embraced 226 
units of the whole circumference, and correspond to 3f degrees. 
Notable is the fact that the Indians never reckoned, like the 
Greeks, with the whole chord of double the arc, but always 
with the sine (joa) and versed sine. Their mode of calculating 
tables was theoretically very simple. The sine of 90** was 
equal to the radius, or 3438; the sine of 30** was evidently 
half that, or 1719. Applying the formula sin*a -f cos*a = r*, 

they obtained sin 45** s-^l^s 2431. Substituting for cos a 
its equal sin (90 — a), and making a = 60**, they obtained 

sin 60** = ^^^= 2978. With the sines of 90, 60, 46, and 30 

as starting-points, they reckoned the sines of half the angles 
by the formula ver sin 2 a = 2 sin^ a, thus obtaining the sines 
of 22* 30', 11** 16', 7** 30', 3** 45'. They now figured out the 
sines of the complements of these angles, namely, the sines 
of 86** 15', 82** 30', 78** 46', 76^ 67** 30' ; then they calculated 
the sines of half these angles; then of their complements; 
then, again, of half their complements ; and so on. By this 
very simple process they got the sines of angles at intervals of 
3** 45'. In this table they discovered the unique law that if 
a, 6, c be three successive arcs such that a — 6 = 6 — c = 3** 46', 

then sin a — sin 6 = (sin h — sin c) — ^^- This formula was 

afterwards used whenever a re-calculation of tables had to 
be made. No Indian trigonometrical treatise on the triangle 



100 A HISTOBY OF MATHEMATICS. 

is extant. In astronomy they solved plane and spherical 
right triangles." 

It is remarkable to what extent Indian mathematics enters 
into the science of our time. Both the form and the spirit 
of the arithmetic and algebra of modem times are essentially 
Indian and not Grecian. Think of' that most perfect of 
mathematical symbolisms — the Hindoo notation, think of the 
Indian arithmetical operations nearly as perfect as our own, 
think of their elegant algebraical methods, and then judge 
whether the Brahmins on the banks of the Ganges are not 
entitled to some credit. Unfortunately, some of the most 
brilliant of Hindoo discoveries in indeterminate analysis 
reached Europe too late to exert the influence they would 
have exerted, had they come two or three centuries earlier. 



THE ARABS. 

After the flight of Mohammed from Mecca to Medina in 
622 A.D., an obscure people of Semitic race began to play an 
important part in the drama of history. Before the lapse of 
ten years, the scattered tribes of the Arabian peninsula were 
fused by the furnace blast of religious enthusiasm into a 
powerful nation. With sword in hand the united Arabs 
subdued Syria and Mesopotamia. Distant Persia and the 
lands beyond, even unto India, were added to the dominions 
of the Saracens. They conquered Northern Africa, and nearly 
the whole Spanish peninsula, but were finally checked from 
further progress in Western Europe by the firm hand of 
Charles Martel (732 a.d.). The Moslem dominion extended 
now from India to Spain; but a war of succession to the 
caliphate ensued, and in 755 the Mohammedan empire was 
divided, — one caliph reigning at Bagdad, the other at Cordova 



THE ARABS. 101 

in Spain. Astounding as was the grand march of conquest by 
the Arabs, still more so was the easfe with which they put 
aside their former nomadic life, adopted a higher civilisation, 
and assumed the sovereignty over cultivated peoples. Arabic 
was made the written language throughout the conquered 
lands. With the rule of the Abbasides in the East began a 
new period in the history of learning. The capital, Bagdad, 
situated on the Euphrates, lay half-way between two old 
centres of scientific thought, — India in the East, and Greece 
in the West. The Arabs were destined to be the custodians 
of the torch of Greek and Indian science, to keep it ablaze 
daring the period of confusion and chaos in the Occident, and 
afterwards to pass it over to the Europeans. Thus science 
passed from Aryan to Semitic races, and then back again 
to the Aryan. The Mohammedans have added but little to 
the knowledge in mathematics which they received. They 
now and then explored a small region to which the path had 
been previously pointed out, but they were quite incapable of 
discovering new fields. Even the more elevated regions in 
which the Hellenes and Hindoos delighted to wander — 
namely, the Greek conic sections and the Indian indeterminate 
analysis — were seldom entered upon by the Arabs. They 
were less of a speculative, and more of a practical turn of 
mind. 

The Abbasides at Bagdad encouraged the introduction of 
the sciences by inviting able specialists to their court, irre- 
spective of nationality or religious belief. Medicine and 
astronomy were their favourite sciences. Thus Haroun-al- 
Raschid, the most distinguished Saracen ruler, drew Indian 
physicians to Bagdad. In the year 772 there came to the 
court of Caliph Almansur a Hindoo astronomer with astronom- 
ical tables which were ordered to be translated into Arabic. 
These tables, known by the Arabs as the Sindhind, and 



102 A HISTORY OF MATHEMATICS. 

probably taken from the Brahma-gphutoraiddhanta of Brahma- 
gupta, stood in great authority. They contained the important 
Hindoo table of sines. 

Doubtless at this time, and along with these astronomical 
tables, the Hindoo numerals, with the zero and the principle 
of position, were introduced among the Saracens. Before the 
time of Mohammed the Arabs had no numerals. Numbers 
were written out in words. Later, the numerous computations 
connected with the financial administration over the conquered 
lands made a short symbolism indispensable. In some locali- 
ties, the numerals of the more civilised conquered nations 
were used for a time. Thus in Syria, the Greek notation was 
retained; in Egypt, the Coptic. In some cases, the numeral 
adjectives may have been abbreviated in writing. The DiwanU 
numercUs, found in an Arabic-Persian dictionary, are supposed 
to be such abbreviations. Gradually it became the practice to 
employ the 28 Arabic letters of the alphabet for numerals, in 
analogy to the Greek system. This notation was in turn 
superseded by the Hindoo notation, which quite early was 
adopted by merchants, and also by writers on arithmetic. Its 
superiority was so universally recognised, that it had no rival, 
except in astronomy, where the alphabetic notation continued 
to be used. Here the alphabetic notation offered no great 
disadvantage, since in the sexagesimal arithmetic, taken from 
the Almagest, numbers of generally only one or two places 
had to be written.' 

As regards the form of the so-called Arabic numerals, the 
statement of the Arabic writer Albiruni (died 1039), who 
spent many years in India, is of interest. He says that the 
shape of the numerals, as also of the letters in India, differed 
in different localities, and that the Arabs selected from the 
various forms the most suitable. An Arabian astronomer 
says there was among people much difference in the use of 



THB AKABS. 108 

symbols, especially of those for 5, 6, 7, and 8. The symbols 
used by the Arabs can be traced back to the tenth century. 
We find material differences between those used by the 
Saracens in the East and those used in the West. But 
most surprising is the fact that the symbols of both the East 
and of the West Arabs deviate so extraordinarily from the 
Hindoo Devatiagari numerals (= divine numerals) of to-day, 
and that they resemble much more closely the apices of 
the Roman writer Boethius. This strange similarity on the 
one hand, and dissimilarity on the other, is difficult to explain. 
The most plausible theory is the one of Woepcke : (1) that'^ 
about the second century after Christ, before the zero had 
been invented, the Indian numerals were brought to Alexan- 
dria, whence they spread to Eome and also to West Africa; 
(2) that in the eighth century, after the notation in India had 
been already much modified and perfected by the invention of 
the zero, the Arabs at Bagdad got it from the Hindoos ; (3) that 
the Arabs of the West borrowed the Columbus-egg, the zero, 
from those in the East, but retained the old forms of the nine 
numerals, if for no other reason, simply to be contrary to their 
])olitical enemies of the East; (4) that the old forms were 
remembered by the West-Arabs to be of Indian origin, and 
were hence called Chibar-numerah (= dust-numerals, in mem- 
ory of the Brahmin practice of reckoning on tablets strewn 
with dust or sand; (5) that, since the eighth century, the 
numerals in India underwent further changes, and assumed 
the greatly modified forms of the modem Devanagari-numer- 
als.* This is rather a bold theory, but, whether true or not, 
it explains better than any other yet propounded, the relations 
between the apices, the Gubar, the East- Arabic, and Devana- ' 
gari numerals. 

It has been mentioned that in 772 the Indian SiddhatUa was 
brought to Bagdad and there translated into Arabic. There 



104 A HISTORY OF MATHEMATICS. 

is no evidence that any intercourse existed between Arabic 
and Indian astronomers either before or after this time, ex- 
cepting the travels of Albiruni. But we should be very slow 
to deny the probability that more extended communications 
actually did take place. 

Better informed are we regarding the way in which Greek 
science, in successive waves, dashed upon and penetrated Arabic 
soil. In Syria the sciences, especially philosophy and medi- 
cine, were cultivated by Greek Christians. Celebrated were the 
schools at Antioch and Emesa, and, first of all, the flourishing 
Nestorian school at Edessa. From Syria, Greek physicians 
and scholars were called to Bagdad. Translations of works 
from the Greek began to be made. A large number of Greek 
manuscripts were secured by Caliph Al Mamun (813-833) from 
the emperor in Constantinople and were turned over to Syria. 
The successors of Al Mamun continued the work so auspic- 
iously begun, until, at the beginning of the tenth century, the 
more important philosophic, medical, mathematical, and as- 
tronomical works of the Greeks could all be read in the Arabic 
tongue. The translations of mathematical works must have 
been very deficient at first, as it was evidently difficult to 
secure translators who were masters of both the Greek and 
Arabic and at the same time proficient in mathematics. The 
translations had to be revised again and again before they 
were satisfactory. The first Greek authors made to speak in 
Arabic were Euclid and Ptolemaeus. This was accomplished 
during the reign of the famous Haroun-al-Raschid. A revised 
translation of Euclid's Elements was ordered by Al Mamun. 
As this revision still contained numerous errors, a new trans- 
lation was made, either by the learned Honein ben Ishak, or 
by his son, Ishak ben Honein. To the thirteen books of the 
Elements were added the fourteenth, written by Hjrpsicles, 
and the fifteenth by Damascius. But it remained for Tabit 



THE ARABS. 105 

ben Korra to bring forth an Arabic Euclid satisfying every 
neecL Still greater difficulty was experienced in securing an 
intelligible translation of the Almagest. Among other impor- 
tant translations into Arabic were the works of Apollonius, 
Archimedes, Heron, and Diophantus. Thus we see that in 
the coarse of one century the Arabs gained access to the vast 
treasures of Greek science. Havi ng been litt le accustomed to 
ab stract t ho ught, we need n ot marvel i f, during the nintK cen- 
tury, all their energy was exhausted merely in appropriating 
the foreigpi matexiaL No attempts were made at original 
work in mathematics until the next century. 

In astronomy, on the other hand, great activity in original 
research existed as early as the ninth century. The religious 
observances demanded by Mohammedapism presented to as- 
tronomers several practical problems. ^The Moslem dominions 
being of such enormous extent, it remained in some localities 
for the astronomer to determine which way the '< Believer" 
must torn during prayer that he may be facing Mecca. The 
prayers and ablutions had to take place at definite hours dur- 
ing the day and night. This led to more accurate determina- 
tions of time. To fix the exact date for the Mohammedan 
feasts it became necessary to observe more closely the motions 
of the moon. In addition to all this, the old Oriental supersti- 
tion that extraordinary occurrences in the heavens in some 
mysterious way affect the progress of human affairs added 
increased interest to the prediction of eclipses.^ 

For these reasons considerable progress was made. Astro- 
nomical tables and instruments were perfected, observatories 
erected, and a connected series of observations instituted. This 
intense love for astronomy and astrology continued during the 
whole Arabic scientific period. As in India, so here, we hardly 
ever find a man exclusively devoted to pure mathematics. Most 
of the so-called mathematicians were first of all astronomers. 



^ 



106 A HISTORY OF MATHEMATICS. 

The first notable author of mathematical books was Moham- 
med ben Musa Hovarezmi^ who lived during the reign of Caliph 
Al Mamun (814--833). He was engaged by the caliph in mak- 
ing extracts from the Sindhind, in revising the tablets of Ptole- 
maeus, in taking observations at Bagdad and Damascus^ and in 
measuring a degree of the earth's meridian. Important to us 
is his work on algebra and arithmetic. The portion on arith- 
metic is not extant in the original^ and it was not till 1857 
that a Latin translation of it was found. It begins thus: 
^^ Spoken has Algoritmi. Let us give deserved praise to Grod, 
our leader and defender." Here the name of the author, Ho- 
varezmi, has passed into Algoritmi^ from which comes pur 
modem word algorithm, signifying the art of computing in 
any particular way. The arithmetic of Hovarezmi, being 
based on the principle of position and the Hindoo method of 
calculation, "excels," says an Arabic writer, "all others in 
brevity and easiness, and exhibits the Hindoo intellect and 
sagacity in the grandest inventions." This book was followed 
by a large number of arithmetics by later authors, which dif- 
fered from the earlier ones chiefly in the greater variety of 
methods. Arabian arithmetics generally contained the four 
operations with integers and fractions, modelled after the 
Indian processes. They explained the operation of casting oxU 
the ffsj which was sometimes called the "Hindoo proof." They 
contained also the regiUa falsa and the regula duorum falsoruniy 
by which algebraical examples could be solved without algebra. 
Both these methods were known to the Indians. iHherjegida 
falsa or falsa positio was the assigning of an assumed value to 
the unknown quantity, which value, if wrong, was corrected 
by some process like the " rule of three." ^jDiophantus used a 
method almost identical with this. The regula duorum fair 
sorum was as follows :' To solve an equation /(a) = F, assume, 
for the moment, two values for x ; namely, » s a and x=zb. 



^ *. 



THE ABABS. 107 

Then form /(a) = A and/(6) = B, and determine the errors 
r-A = ^. and F- B = ^» ; then the required x = ^^* " ^^' 

is generally a close approximation, but is absolutely accurate 
whenever /(a) is a linear function of x. 

We now return to Hovarezmi, and consider the other part 
of his work^ — the algebra. This is the first book known to 
contain this word itself as title. Eeally the title consists of 
two words, aldahebr wdLrnvkabaUx, the nearest English transla- 
tion of which is "restoration" and "reduction." By "res- 
toration" was meant the transposing of negative terms to the 
other side of the equation; by "reduction," the uniting oi/ 
similar terms. Thus, «* — 2aj = oaj-f6 passes by aldshebr into 
aE^s56a; + 2sc + 6; and this, by walmukabala, into a^=7aj + 6. 
The work on algebra, like the arithmetic, by the same author, 
contains nothing original, [it explains the elementary opera- 
tions and the solutions of linear and quadratic equations., 
From whom did the author borrow his knowledge of algebra ? 
That it came entirely from Indian sources is impossible, for 
the Hindoos had no such rules like the "restoration" and 
** reduction." They were, for instance, never in the habit of 
making all terms in an equation positive, as is done by the 
process of " restoration." Diophantus gives two rules which 
resemble somewhat those of our Arabic author, but the proba- 
bility that the Arab got all his algebra from Diophantus is 
lessened by the consideirations that he recognised both roots 
of a quadratic, while Diophantus noticed only one ; and that 
the Greek algebraist, unlike the Arab, habitually rejected irra- 
tional solutions. It would seem, therefore, that the algebra 
of Hovarezmi was neither purely Indian nor purely Greek, 
bat was a hybrid of the two, with the Greek element pre- 
dominating. 

The algebra of Hovarezmi contains also a few meagre 



108 A HISTORY OF MATHEMATICS. 

/fragments on geometry^ He gives the theorem of the right 
triangle, but proves it after Hindoo fashion and only for the 
simplest case, when the right triangle is isosceles. He then 
calculates the areas of the triangle, parallelogram, and circle. -^^ 
For rr he uses the value 3f , and also the two Indian^ ir = VlO 
a^<l ^ = ygj j. Strange to say, the last value was after-r 
wards forgotten by the Arabs, and replaced by others less 
accurate. This bit of geometry doubtless came from India. 
Later Arabic writers got their geometry almost entirely from 
Greece. 

Next to be noticed are the three sons of Musa ben Sakir, 
who lived in Bagdad at the court of the Caliph Al Mamun. 
They wrote several works, of which we mention a geometry 
in which is also contained the well-known formula for the 
area of a triangle expressed in terms of its sides. We are 
told that one of the sons travelled to Greece, probably to 
collect astronomical and mathematical manuscripts, and that 
on his way back he made acquaintance with Tabit ben Korra. 
Eecognising in him a talented and learned astronomer, Mo- 
hammed procured for him a place among the astronomers --at 
the court in Bagdad. Tabit ben Korra (836-901) was born 
at Harran in Mesopotamia. He was proficient not only in 
astronomy and mathematics, but also in the Greek, Arabic, 
and Syrian languages. His translations of ApoUonius, Ar- 
chimedes, Euclid, Ptolemy, Theodosius, rank among the best. 
His dissertation on amicable numbers (of which each is the 
sum of the factors of the other) is the [first known specimen 
of original work in mathematics on Arabic soil.\ It shows 
that he was familiar with the Pythagorean theory of numbers. ' 
Tabit invented the following rule for finding amicable num- 
bers: If i> = 3.2»-l, g = 3.2"-^-l, r = 9.2*»-^ -1 (n being 
a whole number) are three primes, then a = 2^q, 6 = 2"r are 
a pair of amicable numbers. Thus, if n &3 2, then p = 11, 



THE ABAB8. 109 

^ ss 6| r ss 71, and a s 220, b = 284. Tabit also trisected an 
angle. 

Foremost among the astronomers of the ninth century 
ranked Al B attan i, called AJbategnivs by the Latins. Battan 
in Syria was his birthplace. His observations were celebrated 
for great precision. His work, De scientia stellartim, was trans- 
lated into Latin by Plato Tiburtinus, in the twelfth century. 
Out of this translation sprang the word ' sinus/ as the name 
of a trigonometric function. The Arabic word for "sine/' 
dbckibcby was derived from the Sanscrit jiva, and resembled 
the Arabic word dschaib, meaning an indentation or gulf. 
Hence the Latin '^ sinus."' Al Battani was a close student 
of Ptolemy, but did not follow him altogether. He took an 
important step for the better, when he [introduced the Indian 
"sine" or half the chord, in place of the whole chord of 
Ptolemy.^ Another improvement on Greek trigonometry made 
by the Arsba points likewise to Indian influences. Proposi- 
tions and operations which were treated by the Greeks geo- 
metrically are expressed by the Arabs algebraically. Thus, 

Al BcUtani at once gets from an equation ^^^ = D, the value 

^ ^ cos^ 

of 6 by means of sin = — . — a process unknown to 

the ancients. He knows, of course, all the formulas for 
spherical triangles given in the Almagest^ but goes further, 
and adds an important one of his own for oblique-angled 
triangles; namely, cos a =5 cos 6 cose + sin 6 sine cos A 

At the beginning of the tenth century political troubles 
arose in the East, and as a result the house of the Abbasides 
lost powerJ One province after another was taken, till, in 
946. all possessions were wrested from them. Fortunately, 
the Ipew rulers at Bagdad, the Persian Buy idea^ were as much 
interested in astronomy as their predecessors. ^ The progress 
of the sciences was not only unchecked, but the conditions 



110 A HISTOBY OF MATHEBfATIOS. 

for it became even more favourable. The Emir Adud-edrdavla 
(978-983) gloried in having studied astronomy himself. His 
son Saraf-ed-daula erected an observatory in the garden of his 
palace, and called thither a whole group of scholars/ Among 
them were Abut Wefa, Al Kuhi, Al Sagani. 

Abol Wefa (940-998) was bom at Buzshan in Chorassan, a 
region among the Persian mountains, which has brought forth 
many Arabic astronomers. He forms an important exception 
to the unprogressive spirit of Arabian scientists by his brill- 
iant discovery of the variation of the moon, an inequality 
usually supposed to have been first discovered by Tycho 
Brahe.^^ Abul Wefa translated Diophantus. He is one of the 
last Arabic translators and commentators of Greek authors. 
The fact that he esteemed the algebra of Mohammed ben 
Musa Hovarezmi worthy of his commentary indicates that 
thus far algebra had made little or no progress on Arabic soil. 
Abul Wefa invented a/method for computing tables of sines 
which gives the sine ot half a degree correct to nine decimal 
places. J He did himself credit bj^introducing the tangent into 
trigonometry and by calculating a table of tangents. The 
first step toward this had been taken by Al Battani. Unfor- 
tunately, this innovation and the ^scovery of the moon's 
variatioi^'^ excited apparently no. notice among his contempo- 
raries and followers. " We can hardly help looking upon this 
circumstance as an evidence of a servility of intellect belong- 
ing to the Arabian period.'* A treatise by Abul Wefa on 
"geometric constructions" indicates that efforts were being 
made at that time to improve draughting. It contains a neat 
construction of the corners of the regular polyedrons on the 
circumscribed sphere. Here, for the first time, appears the 
condition which afterwards became very famous in the Occi- 
dent, that the construction be effected with a single opening 
of the compass. 



THE ARABS. Ill 

Lai Knhiy the second astronomer at the observatory of the 
emir at Bagdad, was a close student of Archimedes and 
Apollonius. He solved the problem, to construct a segment 
of a sphere equal in volume to a given segment and having 
a curved surface equal in area to that of another given seg- 
ment. He, Al Sagani, and Al Biruni made a study of the 
trisection of angles. Abol Gud, an able geometer, solved the 
problem by the intersection of a parabola with an equilateral 
hyperbola. ^ 

The Arabs had already discovered the theorem that the 
fam of two cubes can never be a cube. Abu Mohammed Al 
Hogoidi of Chorassan thought he had proved this, but we are 
told that the demonstration was defective. Creditable work 
in theory of numbers and algebra was done by Fahri des Al 
Kjffliiy who lived at the beginning of the eleventh century. 
His treatise on algebra is the greatest algebraic work of the 
Arabs. In it he appears as a disciple of Diophantus. He 
was the first to operate with higher roots and to solve equa- 
tions of the form x^ -f oaf = b. For the solution of quadratic 
equations he gives both arithmetical and geometric proofs. 
He was the first Arabic author to give and prove the theorems 
on the summation of the series : — 

SfM-l 

1«4.2' + 3'H hn' = (l + 2H hn)~T', 

1» + 2* + 3» -f ••• + n» = (1 + 2 + ... + n)*. 

Al Karhi also busied himself with indeterminate analysis. 
He showed skill in handling the methods of Diophantus, but 
added nothing whatever to the stock of knowledge already 
on hand. As a subject for original research, indeterminate 
analysis was too subtle for even the most gifted of Arabian 
minds. Rather surprising is the fact that Al Karhi's algebra 
shows no traces whatever of Hindoo indeterminate analysis. 



112 A HISTORY OF MATHEBIATIC8. 

But most astonishing it is^ that an arithmetic by the same 
author completely excludes the Hindoo numerals. It is con- 
structed wholly after Greek pattern. Abul Wefa also, in the 
second half of the tenth century, wrote an arithmetic in which 
Hindoo numerals find no place. This practice is the very 
opposite to that of other Arabian authors. The question, 
why the Hindoo numerals were ignored by so eminent authors, 
is certainly a puzzle. Cantor suggests that at one time there 
may have been rival schools, of which one followed almost 
exclusively Greek mathematics, the other Indian. 

The Arabs were familiar with geometric solutions of quad- 
ratic equations. Attempts were now made to solve cubic 
equations geometrically. They were led to such solutions by 
the study of questions like the Archimedean problem, demand- 
ing the section of a sphere by a plane so that the two seg- 
ments shall be in a prescribed ratio. The first to state this 
problem in form of a cubic equation was Al Mahani of Bagdad, 
while Abu Gafar Al Hazin was the first Arab to solve the 
equation by conic sections. Solutions were given also by 
Al Kuhi, Al Hasan ben Al Haitam, and others.^ Another 
difficult problem, to determine the side of a regular hepta- 
gon, required the construction of the side from the equation 
35* — a^ — 2a;-f-l = 0. It was attempted by many and at last 
solved by Abul Gud. 

The one who did most to elevate to a method the solution 
of algebraic equations by intersecting conies, was Omar al 
Hayyami of Chorassan, about 1079 a.d. He divides cubics into 
two classes, the trinomial and quadrinomial, and each class 
into families and species. Each species is treated separately 
but according to a general plan. He believed that cubics 
could not be solved by calculation, nor bi-quadratics by geom- 
etry. He rejected negative roots and often failed to discover 
all the positive ones. Attempts at bi-quadratic equations 



THE ABABS. 118 

were made by Abul Wef a," who solved geometrically a?* = a 
and a^ + ao* = b. 

JThe solution of cubic equations by intersecting conies was 
the greatest achievement of the Arabs in algebra.^ The foun- 
dation to this work had been laid by the Greeks, for it was 
Mensechmus who first constructed the roots of a^ — a = or 
ac" — 2a'=0. It was not his aim to find the number corre- 
sponding to X, but simply to determine the side a; of a cube 
double another cube of side a. The Arabs, on the other 
hand, had another object in view : to find the roots of given 
numerical equations. In the Occident, the Arabic solutions 
of cubics remained unknown until quite recently. Descartes 
and Thomas Baker invented these constructions anew. The 
works of Al Hayyami, Al Karhi, Abul Gud, show how the 
Arabs departed further and further from the Indian methods, 
and placed themselves more immediately under Greek influ- 
ences. In this way they barred the road of progress against 
themselves. The Greeks had advanced to a point where 
material progress became difficult with their methods ; but the 
Hindoos furnished new ideas, many of which the Arabs now 
rejected. 

With Al Karhi and Omar Al Hayyami, mathematics among 
the Aiabs of the East reached flood-mark, and now it begins to 
ebb. Between 1100 and 1300 a.d. come the crusades with 
war and bloodshed, during which European Christians profited 
much by their contact with Arabian culture, then far superior 
to their own; but the Arabs got no science from the Christians 
in return. The crusaders were not the only adversaries of the 
Arabs. During the first half of the thirteenth century, they 
had to encounter the wild Mongolian hordes, and, in 1256, were 
conquered by them under the leadership of Hulagu. The 
caliphate at Bagdad now ceased to exist. At the close of the 
fourteenth century still another empire was formed by Timur 



114 A HISTORY OF MATHEMATICS. 

or Tamerlane, the Tartar. During such sweeping turmoil, it 
is not surprising that science declined. Indeed, it is a marvel 
that it existed at all. During the supremacy of Hulagu, lived 
Nasir Eddin (1201-1274), a man of broad culture and an able 
astronomer. He persuaded Hulagu to build him and his asso- 
ciates a large observatory at Maraga. Treatises on algebra, 
geometry, arithmetic, and a translation of Euclid's Elements^ 
were prepared by him. Even at the court of Tamerlane in 
Samarkand, the sciences were by no means neglected. A 
group of astronomers was drawn to this court. Ulug Beg 
(1393-1449), a grandson of Tamerlane, was himself an 
astronomer. Most prominent at this time was Al Kaschi, the 
author of an arithmetic. Thus, during intervals of peace, 
science continued to be cultivated in the East for several 
centuries. The last Oriental writer was Beha Eddin (1547- 
1622). His Essence of Arithmetic stands on about the same 
level as the work of Mohammed ben Musa Hovarezmi, written 
nearly 800 years before. 

"Wonderful is the expansive power of Oriental peoples, 
with which upon the wings of the wind they conquer half 
the world, but more wonderful the energy with which, in 
less than two generations, they raise themselves from the 
lowest stages of cultivation to scientific efforts." During 
all these centuries, astronomy and mathematics in the Orient 
greatly excel these sciences in the Occident. 

Thus far we have spoken only of the Arabs in the East. 
Between the Arabs of the East and of the West, which were 
under separate governments, there generally existed consider- 
able political animosity. In consequence of this, and of the 
enormous distance between the two great centres of learning, 
Bagdad and Cordova, there was less scientific intercourse 
among them than might be expected to exist between peoples 
having the same religion and written language. Thus the 



THE ABABS. 115 

coarse of science in Spain was quite independent of that in 
Persia. While wending our way westward to Cordova, we 
must stop in Egypt long enough to observe that there, too, 
scientific activity was rekindled. Not Alexandria, but Cairo 
with its library and observatory, was now the home of learn- 
ing. Foremost among her scientists ranked Ben Janus (died 
1008), a contemporary of Abul Wefa. He solved some difficult 
problems in spherical trigonometry. Another Egyptian astron- 
omer was Ibn Al Haitam (died 1038), who wrote on geometric 
locL Travelling westward, we meet in Morocco Abal Hasan 
All, whose treatise ' on astronomical instruments ' discloses a 
thorough knowledge of the Conies of ApoUonius. Arriving 
finally in Spain at the capital, Cordova, we are struck by the 
magnificent splendour of her architecture. At this renowned 
seat of learning, schools and libraries were founded during the 
tenth century. 

Little is known of the progress of mathematics in Spain. 
The earliest name that has come down to us is Al Madshriti 
(died 1007), the author of a mystic paper on * amicable num- 
bers.' His pupils founded schools at Cordova, Dania, and 
Granada. But the only great astronomer among t he Sarac ens 
in Spain is Gabir ben Aflah of Sevilla, frequently called Oeber, 
He lived in the second half of the eleventh century. It was 
formerly believed that he was the inventor of algebra, and that 
the word algebra came from * Gabir' or *Geber.' He ranks 
among the most eminent astronomers of this time, but, like so 
many of his contemporaries, his writings contain a great deal 
of mysticism. His chief work is an astronomy in nine books, of 
which the first is devoted to trigonometry. In his treatment 
of spherical trigonometry, he exercises great independence of 
thought. He makes war against the time-honoured procedure 
adopted by Ptolemy of applying " the rule of six quantities," 
and gives a new way of his own, based on the ^ rule of four 



116 A HISTORY OF MATHEMATICS. 

quantities.' This is : If PPi and QQi be two arcs of gpreat 
circles intersecting in A^ and if PQ and PiQi be arcs of great 
circles drawn perpendicular to QQ^^ then we have the propor- 
tion 

sin AP : sin PQ = sin APi : sin PiQy. 

From this he derives the formulas for spherical right triangles. 
To the four fundamental formulas already given by Ptolemy, 
he added a fifth, discovered by himself. If a, 5, c, be the sides, 
and A, B, C, the angles of a spherical triangle, right-angled at 
A, then cos 5 = cos 6 sin C. This is frequently called " Geber's 
Theorem." Eadical and bold as were his innovations in 
spherical trigonometry, in plane trigonometry he followed 
slavishly the old beaten path of the Greeks. Not even did he 
adopt the Indian ' sine ' and * cosine,' but still used the Greek 
< chord of double the angle.' So painful was the departure 
from old ideas, even to an independent Arab ! After the time 
of Gabir ben Aflah there was no mathematician among the 
Spanish Saracens of any reputation. In the year in which 
Columbus discovered America, the Moors lost their last foot- 
hold on Spanish soil. 

We have witnessed a laudable intellectual activity among 
the Arabs. They had the good fortune to possess rulers 
who, by their munificence, furthered scientific research. At 
the courts of the caliphs, scientists were supplied with libra- 
ries and observatories. A large number of astronomical and 
mathematical works were ¥rritten by Arabic authors. Yet 
we fail to find a single important principle in mathematics 
brought forth by the Arabic mind. Whatever discoveries 
they made, were in fields previously traversed by the Greeks 
or the Indians, and consisted of objects which the latter had 
overlooked in their rapid march. The Arabic mind did not 
possess that penetrative insight and invention by which mathe- 
maticians in Europe afterwards revolutionised the science. 



BUROFE DURING THE MIDDLE AGES. 117 

The Arabs were learned^ but not origiual. Their chief service 
to science consists in this, that they adopted the learning of 
Greece and India, and kept what they received with scrupu- 
lous care. When the love for science began to grow in the 
Occident, they transmitted to the Europeans the valuable 
treasures of antiquity. ^Thus a Semitic race was, during the 
Dark Ages, the custodian of the Aryan intellectual possessions!} 



EUKOPE DUKING THE MIDDLE AGES. 

With the third century after Christ begins an era of migra- 
tion of nations in Europe. The powerful Groths quit their 
swamps and forests in the North and sweep onward in steady 
soathwestem current, dislodging the Vandals, Sueves, and 
Burgundians, crossing the Eoman territory, and stopping and 
recoiling only when reaching the shores of the Mediterranean. 
From the Ural Mountains wild hordes sweep down on the 
Danube. The Roman Empire falls to pieces, and the Dark 
Ages begin. But dark though they seem, they are the germi- 
nating season of the institutions and nations of modem Europe. 
The Teutonic element, partly pure, partly intermixed with the 
Celtic and Latin, produces that strong and luxuriant growth, 
the modem civilisation of Europe. Almost all the various 
nations of Europe belong to the Aryan stock. As the Greeks 
and the Hindoos — both Aryan races — were the great thinkers 
of antiquity, so the nations north of the Alps became the great 
intellectual leaders of modern times. 

Introduction of Soman Matfiematics, 

We shall now consider how these as yet barbaric nations of 
the North gradually came in possession of the intellectual 



118 A HISTORY OF MATHEMATICS. 

treasures of antiquity. With the spread of Christianity the 
Latin language was introduced not only in ecclesiastical but 
also in scientific and all important worldly transactions. Nat- 
urally the science of the Middle Ages was drawn largely from 
Latin sources. In fact, during the earlier of these ages Ro- 
man authors were the only ones read in the Occident. Though 
Greek was not wholly unknown, yet before the thirteenth 
century not a single Greek scientific work had been read or 
translated into Latin. Meagre indeed was the science which 
could be gotten from Eoman writers, and we must wait several 
centuries before any substantial progress is made in mathe- 
matics. 

After the time of Boethius and Cassiodorius mathematical 
activity in Italy died out. tThe first slender blossom of science 
among tribes that came from the North was an encyclopaedia 
entitled Origines, written by Isidorus (died 636 as bishop of 
Seville)^ This work is modelled after the Roman encyclopae- 
dias of Martianus Capella of Carthage and of Cassiodorius. 
Part of it is devoted to the quadrivium, arithmetic, music, 
geometry, and astronomy. He gives definitions and grammat- 
ical explications of technical terms, but does not describe the 
modes of computation then in vogue. After Isidorus there 
follows a century of darkness which is at last dissipated by 
the appearance of Bede the Venerable (672-735), the most 
learned man of his tiiSe^ He was a native of Ireland, then 
the home of learning in the Occident. His works contain 
treatises on the ComptUiiSy or the computatiou of Easter-timeu 
and on finger-reckoning. It appears that a .finger-symbolism; 
was then widely used for calculation. The correct determina- 
tion of the time of Easter was a problem which in those days 
greatly agitated the Church. It became desirable to have at 
least one monk at each monastery who could determine the 
day of religious festivals and could compute the calendar. 



EUROPE DUKI^G THE MIDDLE AGES. 119 

Such deterininations required some knowledge of arithmetic. 
Hence we find that the art of calculating always found some 
little corner in the curriculum for the education of monks. 

The year in which Bede died is also the year in which 
Alcnin (735-804) was bom. Alcuin was educated in Ireland, 
and was called to the court of Charlemagne to direct the prog- 
ress of education in the great Frankish Empire. Charlemagne 
^^ a great patron of learning and of learned men. In the 
great sees and monasteries he founded schools in which were 
taught the psalms, writing, singing, computation (computiLs), 
and grammar. By computus was here meant, probably, not 
naeiely the determination of Easter-time, but the art of com- 
putation in general. Exactly what modes of reckoning were 
then employed we have no means of knowing. It is not likely 
that Alcuin was familiar with the apices of Boethius or with 
the Soman method of reckoning on the abacus. He belongs 
to that long list of scholars who dragged the theory of num- 
^^8 into theology. Thus the number ot beings created by 
^^^ who created all things well, is 6, because 6 is a perfect 
'^^inber (the sum of its divisors being 1 -j- 2 -f3 = 6) ; 8, on 
*^e other hand, is an imperfect number (1 -f 2 -f 4 < 8) ; hence 
*^^ second origin of mankind emanated from the number 8, 
^ch is the number of souls said to have been in Noah's ark. 
There is a collection of "Problems for Quickening the 
^ind" {proponUones ad actiendos iuvenes), which are certainly 
r^ old as 1000 A.D. and possibly older. Cantor is of the opin- 
^^"H that they were written much earlier and by Alcuin. The 
^^llowing is a specimen of these " Problems " : A dog chasing 
^ rabbit, which has a start of 160 feet, jumps 9 feet every time 
^^e rabbit jumps 7. In order to determine in how many leaps 
^lie dog overtakes the rabbit, 150 is to be divided by 2. In 
^liis collection of problems, the areas of triangular and quad- 
rangular pieces of land are found by the same formulas of 



120 A HISTORY OF MATHEMATICS. 

approximation as those used by the Egyptians andgiven by 
Boethius in his geometiy. An old problem is the V* cistern- 
problem "J (given the time in which several pipes can fill a 
cistern singly, to find the time in which they fill it jointly), 
which has been found previously in Heron,* in the Greek An^ 
thology, and in Hindoo works. Many, of the problems show that 
the collection was compiled chiefly from Boman sources. The 
problem which, on account of its uniqueness, gives the most 
positive testimony regarding the Boman origin is that on the 
interpretation of a will in a case where twins are bom. The 
problem is identical with the Boman, except that different 
ratios are chosen. Of the exercises for recreation, we mention 
the one of the wolf, goat, and cabbage, to be rowed across a 
river in a boat holding only one besides the ferry-man. Query : 
How must he carry them across so that the goat shall not eat 
the cabbage, nor the wolf the goat? The solutions of the 
"problems for quickening the mind" require no further knowl- 
edge than the recollection of some few formulas used in 
surveying, the ability to solve linear equations and to per- 
form the four fundamental operations with integers. Ex- 
traction of roots was nowhere demanded; fractions hardly 
ever occur.' 

The great empire of Charlemagne tottered and fell almost 
immediately after his death. War and confusion ensued. 
Scientific pursuits were abandoned, not to be resumed until 
the close of the tenth century, when under Saxon rule in Ger- 
many and Capetian in France, more peaceful times began. 
The thick gloom of ignorance commenced to disappear. The 
zeal with which the study of mathematics was now taken up 
by the monks is due principally to the energy and influence 
of one man, — Gerbert. He was bom in Aurillac in Auvergne. 
After receiving a monastic education, he engaged in study, 
chiefly of mathematics, in Spain. On his return he taught 



EUROPE DURING THE MIDDLE AGES. 121 

school at Eheims for ten years and became distinguished for 
his profound scholarship. By King Otto I. and his successors 
Grerbert was held in highest esteem. He was elected bishop 
of KheimSy then of Ravenna, and finally was made Pope under 
the name of Sylvester II. by his former pupil Emperor Otho 
XXL He died in 1003, after a life intricately involved in many 
political and ecclesiastical quarrels. Such was the career of 
the greatest mathematician of the tenth century in Europe. 
By his contemporaries his mathematical knowledge was con- 
sidered wonderf uL Many even accused him of criminal inter- 
coarse with evil spirits. 

Gerbert enlarged the stock of his knowledge by procuring 
copies of rare books. Thus in Mantua he found the geometry 
of Boethius. Though this is of small scientific value, yet it 
is of great importance in history. It was at that time the 
only book from which European scholars could learn the ele- 
ments of geometry. Gerbert studied it with zeal, and is 
generally believed himself to be the author of a geometry. 
H. Weissenbom denies his authorship, and claims that the 
book in question consists of three parts which cannot come 
from one and the same author.*^ This geometry contains 
nothing more than the one of Boethius, but the fact that 
occasional errors in the latter are herein corrected shows that 
the author had mastered the subject. '' The first mathemat- 
ical paper of the Middle Ages which deserves this name," 
says Hankel, ''is a letter of Gerbert to Adalbold, bishop of 
Utrecht," in which is £xplained~^the reason why the area of a 
triangle, obtained " geometrically " by taking the product of 
the base by half its altitude, differs from the area calculated 
" arithmetically," according to the formula ^a (a-^- 1), used 
by surveyors/jwhere a stands for a side of an equilateral tri- 
angle. He gives the correct explanation that in the latter 
formula all the small squares, in which the triangle is sup- 



122 A HISTORY OF MATHEMATICS. 

posed to be divided, are counted in wholly, even though parts 
of them project beyond it. 

Gerbert made a careful study of the arithmetical works of 
Boethius. He himself published two works, — RtUe of Com- 
putation on the AbacuSy and A SmaU Book on the Division of 
Numbers. They give an insight into the methods of calcu- 
lation practised in Europe before the introduction of the 
Hindoo numerals. Gerbert used the abacus, which was prob- 
ably unknown to Alcuin. Bemelinus, a pupil of Grerbert, 
describes it as consisting of a smooth board upon which geome- 
tricians were accustomed to strew blue sand, and then to draw 
their diagrams. For arithmetical pur^bses the board was 
divided into 30 columns, of which 3 were reserved for frac- 
tions, while the remaining 27 were divided into groups with 
3 columns in each. In every group the columns were marked 
respectively by the letters C (centum), D (decem), and 
S (singylaris) or M (monas). Bernelinus gives the nine 
numerals used, which are the apices of Boethius, and then 
remarks that the Greek letters may be used in their place.* 
By the use of these columns any number can be written 
without introducing a zero, and all operations in arithmetic 
can be performed in the same way as we execute ours without 
the columns, but with the symbol for zero. Indeed, the 
methods of adding, subtracting, and multiplying in vogue 
among the abacists agree substantially with those of to-day. 
But in a division there is very great difference. (^Tlie early rules 
for division appear to have been framed to satisfy the following 
three conditions : (1) The use of the multiplication table shall 
be restricted as far as possible; at least, it shall never be 
required to multiply mentally a figure of two digits by another 
of one digit. (2) Subtractions shall be avoided as much as 
possible and replaced by additions. (3) The operation shall 

proceed in a purely mechanical way, without requiring trials/ 7 

•—J 



£DBOPB DDBIKO THE MIDDLE AGES. 



That it should be necessary to make such conditions seems 
strange to lu ; but it muBt be remembered that the monks of 
the Middle Ages did not attend school during childhood and 
learn the multiplication table while the memory was fresh. 
Gerbert's rules for division are the oldest extant. They are 
BO brief aa to be very obscure to the uninitiated. They were 
probably intended simply to aid the memory by calling to 
mind the successive steps in the work. la later manuscripts 
they are stated more fully. In dividing any number by another 
of one digit, say 668 by 6, the divisor was first increased to 10 
by adding 4. The process is exhibited in the adjoining figure.* 
Ab it continues, we must imagine the digits 
which are crossed out, to be erased and then 
replaced by the ones beneath. It is as follows : 
600 ->- 10 = 60, but, to rectify the error, 4 x 60, 
or 240, must be added ; 200 ->- 10 = 20, but 4 x 20, 
or 80, must be added. We now write for 
60 + 40 + 80, its sum 180, and continue thus : 
100 -I- 10 = 10; the correction necessary is 4 x 10, 
or 40, which, added to 80, gives 120. Now 
100 -•- 10 = 10, and the correction 4 x 10, to- 
gether with the 20, gives 60. Proceeding as 
before, 60 + 10 = 6 ; the correction is 4 x 6 = 24. 
Now 20 -I- 10 = 2, the correction being 4x2 = 8. 
In the column of units we have now 8 + 4+8, 
or 20. As before, 20-»-10 = 2; the correction 
ifl 2 X 4 = 8, which is not divisible by 10, but 
only by 6, giving the quotient 1 and the re- 
tnainder 2. All the partial quotients taken 
togeUier give 60 + 20 + 10 + 10 + 6 + 2 + 2 + 1 = 111, and 
the remainder 2. 

Similar but more complicated, is the process when the 
divisor oontaios two or more digits. Were the divisor 27, 



CDS 
6 
i 

6 8 

"7 a 7" 
^ * i 

1 p ? 

J i ? 

% 2 

i 

^__ 



124 A HISTORY OF MATHEBCATICS. 

then the next higher multiple of 10, or 30, would be taken 
for the divisor, but corrections would be required for the 3. 
He who has the patience to carry such a division through 
to the end, will understand why it has been said of Gerbert 
that <<Eegulas dedit, quae a sudantibus abacistis vix intelli- 
guntur." He will also perceive why the Arabic method of 
division, when first introduced, was called the divisio aureOj 
but the one on the abacus, the divisio ferrea. 

In his book on the abacus, Bemelinus devotes a chapter to 
fractions. These are, of course, the duodecimals^ first used 
by the Eomans. For want of a suitable notation, calculation 
with them was exceedingly difficult. It would be so even to 
us, were we accustomed, like the early abacists, to express 
them, not by a numerator or denominator, but by the appli- 
cation of names, such as uncia for -j^, quincutk^ for ^ dodrans 
for^. 

In the tenth century, Grerbert was the central figure among 
the learned. In his time the Occident came into secure posses- 
sion of all mathematical knowledge of the Eomans. During 
the eleventh century it was studied assiduously. Though 
numerous works were written on arithmetic and geometry, 
mathematical knowledge in the Occident was still very insig- 
nificant. Scanty indeed were the mathematical treasures 
obtained from Boman sources. 

Translation of Arabic Maniiscripts, 

By his great erudition and phenomenal activity, Gerbert 
infused new life into the study not only of mathematics, but 
also of philosophy. Pupils from France, Germany, and Italy 
gathered at Rheims to enjoy his instruction. When they 
themselves became teachers, they taught of course not only 
the use of the abacus and geometry, but also what they had 



SUBOFE DUBING THE MIDDLE AGES. 12$ 

learned of the philosophy of Aristotle. His philosophy was 
known, at first, only through the writings of Boethius. But 
the growing enthusiasm for it created a demand for his com- 
plete works. Greek texts were wanting. But the Latins 
heard that the Arabs, too, were great admirers of Peripatetism, 
and that they possessed translations of Aristotle's works and 
commentaries thereon. This led them finally to search for 
and translate Arabic manuscripts. During this search, mathe- 
matical works also came to their notice, and were translated 
into Latin. Though some few unimportant works may have 
been translated earlier, yet the period of greatest activity 
began about 1100. The zeal displayed in acquiring the 
Mohammedan treasures of knowledge excelled even that of 
the Arabs themselves, when, in the eighth century, they 
plundered the rich coffers of Greek and Hindoo science. 

Among the earliest scholars engaged in translating manu- 
scripts into Latin was Athelard of Bath. The period of his 
activity is the first quarter of the twelfth century. He 
travelled extensively in Asia Minor, Egypt, and Spain, and 
braved a thousand perils, that he might acquire the language 
and science of the Mohammedans. He made the earliest 
translations, from the Arabic, of Euclid's Elements and of 
the astronomical tables of Mohammed ben Musa Hovarezmi. 
In 1857, a manuscript was found in the library at Cambridge, 
which proved to be the arithmetic by Mohammed ben Musa 
in Latin. This translation also is very probably due to 
Athelard. 

At about the same time flourished Plato of Tivoli or Plato 
Tiburtinus, He effected a translation of the astronomy of 
Al Battani and of the SpluBrica of Theodosius. Through the 
former, the term sinus was introduced into trigonometry. 

About the middle of the twelfth century there was a group 
of Christian scholars busily at work at Toledo, under the 



126 A HISTORY OF MATHEMATICS. 

leadership of Raymond^ then archbishop of Toledo. Among 
those who worked under his direction, John of Seville was 
most prominent. He translated works chiefly on Aristotelian 
philosophy. Of importance to us is a liber algorismiy com- 
piled by him from Arabic authors. On comparing works like 
this with those of the abacists, we notice at once the most 
striking difference, which shows that the two parties drew 
from independent sources. It is argued by some that Ger- 
bert got his apices and his arithmetical knowledge, not from 
Boethius, but from the Arabs in Spain, and that part or the 
whole of the geometry of Boethius is a forgery, dating from 
the time of Gerbert. If this were the case, then the writings 
of Gerbert would betray Arabic sources, as do those of John 
of Seville. But no points of resemblance are found. Gerbert 
could not have learned from the Arabs the use of the abacus, 
because all evidence we have goes to show that they did not 
employ it. Nor is it probable that he borrowed from the 
Arabs the apices, because they were never used in Europe 
except on the abacus. In illustrating an example in division, 
mathematicians of the tenth and eleventh centuries state an 
example in Eoman numerals, then draw an abacus and insert 
in it the necessary numbers with the apices. Hence it seems 
probable that the abacus and apices were borrowed from the 
same source. The contrast between authors like John of 
Seville, drawing from Arabic works, and the abacists, consists 
in this, that, unlike the latter, the former mention the Hin- 
doos, use the term algorism, calculate with the zero, and do 
not employ the abacus. The former teach the extraction of 
roots, the abacists do not; they teach the sexagesimal frac- 
tions used by the Arabs, while the abacists employ the duo- 
decimals of the Romans.^ 

A little later than John of Seville flourished Gerard of 
Cremona in Lombardy. Being desirous to gain possession of 



EUBOFB DIJBINO THE MIDDLB AOB8. 127 

the Almagest, he went to Toledo^ and there, in 1175, translated 
this great work of Ptolemy. Inspired by the richness of 
Mohammedan literature, he gave himself up to its study. He 
translated into Latin over 70 Arabic works. Of mathematical 
treatises, there were among these, besides the Almagest, the 
15 books of Euclid, the Sphcerica of Theodosius, a work of 
Menelaus, the algebra of Mohammed ben Musa Hovarezmi, the 
astronomy of Dshabir ben Aflah, and others less important. 

In the thirteenth century, the zeal for the acquisition of 
Arabic learning continued. Foremost among the patrons of 
science at this time ranked Emperor Frederick II. of Hohen- 
staufen (died 1250). Through frequent contact with Mo- 
hammedan scholars, he became familiar with Arabic science. 
He employed a number of scholars in translating Arabic 
manuscripts, and it was through him that we came in posses- 
sion of a new translation of the Almagest. Another royal 
head deserving mention as a zealous promoter of Arabic 
science was Alfonso X. of Castile (died 1284). He gathered 
around him a number of Jewish and Christian scholars, who 
translated and compiled astronomical works from Arabic 
sources. Rabbi Zag and lehuda ben Mose Cohen were the 
most prominent among them. Astronomical tables prepared 
by these two Jews spread rapidly in the Occident, and con- 
stituted the basis of all astronomical calculation till the 
sixteenth century.' The number of scholars who aided in 
transplanting Arabic science upon Christian soil was large. 
But we mention only one more. Giovanni Campano of Novara 
(about 1260) brought out a new translation of Euclid, which 
drove the earlier ones from the field, and which formed the 
basis of the printed editions.' 

At the close of the twelfth century, the Occident was in 
possession of the so-called Arabic notation. The Hindoo 
methods of calculation began to supersede the cumbrous meth- 



128 A HISTORY OF MATHEMATICS. 

ods inherited from Borne. Algebra, with its rules for solving 
linear and quadratic equations, had been made accessible to 
the Latins. The geometry of Euclid, the SpJioerica of Theodo- 
sius, the astronomy of Ptolemy, and other works were now 
accessible in the Latin tongue. Thus a great amount of new 
scientific material had come into the hands of the Christians. 
The talent necessary to digest this heterogeneous mass of 
knowledge was not wanting. The figure of Leonardo of Pisa 
adorns the vestibule of the thirteenth century. 

It is important to notice that no work either on mathematics 
or astronomy was translated directly from the Greek previous 
to the fifteenth century. 

The First Awakening and its Seqtiel. 

Thus far, France and the British Isles have been the head- 
quarters of mathematics in Christian Europe. But at the 
beginning of the thirteenth century the talent and activity 
of one man was sufiicient to assign the mathematical science 
a new home in Italy. This man was not a monk, like Bede, 
Alcuin, or Grerbert, but a merchant, who in the midst of 
business pursuits found time for scientific study. )^ Leonardo 
of Pisajs the man to whom we owe the first renaissance of 
mathematics on Christian soil. He is also called Fibonacci, 
i.e. son of Bonaccio. His father was secretary at one of the 
numerous factories erected on the south and east coast of the 
Mediterranean by the enterprising merchants of Pisa. He 
made Leonardo, when a boy, learn the use of the abacus. The 
boy acquired a strong taste for mathematics, and, in later years, 
during his extensive business travels in Egypt, Syria, Greece, 
and Sicily, collected from the various peoples all the knowl- 
edge he could get on this subject. Of all the methods of 
calculation, he found the Hindoo to be unquestionably the 



EUROPE DITRING THE MIDDLE AGES. 129 

best. Betarning to Pisa, he published, in 1202, his great 
work, the ^Gber Abaci. 'A revised editionjof this appeared in 
1228. This work contains about all the knowledge the Arabs 
possessed in arithmetic and algebra, and treats the subject in 
a free and independent way. This, together with the other 
books of Leonardo, shows that he was not merely a compiler, 
or, like other writers of the Middle Ages, a slavish imitator 
of the form in which the subject had been previously pre- 
sented, but that he was an original worker of exceptional 
power. 

He was the first great mathematician to advocate the adop- 
tion of the '< Arabic notation." The calculation with the zero 
was the portion of Arabic mathematics earliest adopted by 
the Christians. The minds of men had been prepared for the 
reception of this by the use of the abacus and the apices. 
The reckoning with columns was gradually abandoned, and 
the very word dbactis changed its meaning and became a 
synonym for algorism. For the zero, the Latins adopted 
the name zephirum, from the Arabic sifr («t/ra= empty) ; 
hence our English word cipher. The new notation was 
accepted readily by the enlightened masses, but, at first, 
rejected by the learned circles. The merchants of Italy used 
it as early as the thirteenth century, while the monks in the 
monasteries adhered to the old forms. In 1299, nearly 100 
years after the publication of Leonardo's Liber Abaci, the 
Florentine merchants were forbidden the use of the Arabic 
numerals in book-keeping, and ordered either to employ the 
Boman numerals or to write the numeral adjectives out in 
fulL In the fifteenth century the abacus with its counters 
ceased to be used in Spain and Italy. In France it was used 
later, and it did not disappear in England and Germany before 
the middle of the seventeenth century." Thus, in the Winter^ s 
Tale (iv. 3), Shakespeare lets the clown be embarrassed by 



180 A HISTORY OF MATHEMATIG8. 

a problem which he could not do without counters. lago 
(in Othello, i. 1) expresses his contempt for Michael Gasso^ 
'' forsooth a great mathematician," by calling him a '' counter- 
caster." So general, indeed, says Peacock, appears to have 
been the practice of this species of arithmetic, that its rules 
and principles form an essential part of the arithmetical 
treatises of that day. The real fact seems to be that the old 
methods were used long after the Hindoo numerals were in 
common and general use. With such dogged persistency does 
man cling to the old ! 

The Liber Abaci was, for centuries, the storehouse from 
which authors got material for works on arithmetic and 
algebra. In it are set forth the most perfect methods of 
calculation with integers and fractions, known at that time; 
the square and cube root are explained; equations of the first 
and second degree leading to problems, either determinate 
or indeterminate, are solved by the methods of 'single' or 
'double position,' and also by real algebra. The book con- 
tains a large number of problems. The following was pro- 
posed to Leonardo of Pisa by a magister in Constantinople, 
as a difficult problem : If A gets from B 7 denare, then A's 
sum is five-fold B's ; if B gets from A 6 denare, then B's sum 
is seven-fold A's. How much has each? The Liber Abaci 
contains another problem, which is of historical interest, 
because it was given with some variations by Ahmes, 3000 
years earlier : 7 old women go to Rome ; each woman has 
7 mules, each mule carries 7 sacks, each sack contains 7 loaves, 
with each loaf are 7 knives, each knife is put up in 7 sheaths. 
What is the sum total of all named? Ana, 137,256.* 

In 1220, Leonardo of Pisa published his Practica Geometrtce, 
which contains all the knowledge of geometry and trigonom- 
etry transmitted to him. The writings of Euclid and of some 
other Greek masters were known to him, either from Arabic 



EUROPE DURING THE MIDDLE AGES. ISl 

manuscripts directly or from the translations made by his 
countrymen, Gerard of Cremona and Plato of Tivoli. Leo- 
nardo's Geometry contains an elegant geometrical demon- 
stration of Heron's formula for the area of a triangle, as a 
function of its three sides. Leonardo treats the rich material 
before him with skill and Euclidean rigour. 

Of still greater interest than the preceding works are those 
containing Fibonacci's original investigations. We must here 
preface that after the publication of the Liber Abaci, Leonardo 
was presented by the astronomer Dominicus to Emperor 
Frederick II. of Hohenstaufen. On that occasion, John of 
Palermo, an imperial notary, proposed several problems, which 
Leonardo solved promptly. The first problem was to find a 
number x, such that a? -{-5 and a^ — 5 are each square numbers. 
The answer is a = 3^; for {3^y -{• 5 = {^y, (S^y^5 
s= (2^y, His masterly solution of this is given in his liber 
quadrcUorum, a copy of which work was sent by him to 
Frederick II. The problem was not original with John of 
Palermo, since the Arabs had already solved similar ones. 
Some parts of Leonardo's solution may have been borrowed 
from the Arabs, but the method which he employed of building 
squares by the summation of odd numbers is original with him. 

The second problem proposed to Leonardo at the famous 
scientific tournament which accompanied the presentation of 
this celebrated algebraist to that great patron of learning, 
Emperor Frederick II., was the solving of the equation 
a^ -I- 20^*4- 10 a: = 20. As yet cubic equations had not been 
solved algebraically. Instead of brooding stubbornly over 
this knotty problem, and after many failures still entertaining 
new hopes of success, he changed his method of inquiry and 
showed by clear and rigorous demonstration that the roots 
of this equation could not be represented by the Euclidean 
irrational quantities, or, in other words, that they could not be 



182 A HISTORY OF MATHEMATICS. 

constructed with the ruler and compass only. He contented 
himself with finding a very close approximation to the 
required root. His work on this cubic is found in the Flos, 
together with the solution of the following third problem 
given him by John of Palermo: Three men possess in com- 
mon an unknown sum of money i; the share of the first 

is -; that of the second, -; that of the third, -. Desirous of 
2 3 6 

depositing the sum at a safer place, each takes at hazard a 
certain amount; the first takes x, but deposits only -; the sec- 
ond carries y, but deposits only |; the third takes z, and 
deposits -• Of the amount deposited each one must receive 

exactly ^, in order to possess his share of the whole sum. 
Find Xy Iff z. Leonardo shows the problem to be indetermi- 
nate. Assuming 7 for the sum drawn by each from the 
deposit, he finds i = 47, a? = 33, y = 13, « = 1. 

One would have thought that after so brilliant a beginning, 
the sciences transplanted from Mohammedan to Christian soil 
would have enjoyed a steady and vigorous development. But 
this was not the case. ^During the fourteenth and fifteenth 
centuries, the mathematical science was almost stationary. 
Long wars absorbed the energies of the people and thereby 
kept back the growth of the sciences. The death of Frederick 
II. in 1264 was followed by a period of confusion in Germany. 
The German emperors and the popes were continually quarrel- 
ling, and Italy was inevitably drawn into the struggles between 
the Guelphs and the Ghibellines. France and England were 
engaged in the Hundred Years' War (1338-1453). Then 
followed in England the Wars of the Roses. CThe growth of 
science was retarded not only by war, but also by the inju- 
rious influence of scholastic philosophy.J^ The intellectual 
leaders of those times quarrelled over subtle subjects in meta- 



SUBOPB DXTRING THB MIDDLE AQB8. ISS 

physics and theology. Frivolous questions, such as ''How 
many angels can stand on the point of a needle?" were dis- 
cussed with great interest. Indistinctness and confusion of 
ideas characterised the reasoning during this period. Among 
the mathematical productions of the Middle Ages, the works 
of Leonardo of Pisa appear to us like jewels among quarry- 
rubbish. The writers on mathematics during this period were 
not few in number, but their scientific efforts were vitiated 
by the method of scholastic thinking. Though they possessed 
the Elements of Euclid, yet the true nature of a mathematical 
proof was so little understood, that Hankel believes it no 
exaggeration to say that '' since Fibonacci, not a single proof, 
not borrowed from Euclid, can be found in the whole literature 
of these ages, which fulfils all necessary conditions." 

The only noticeable advance is a simplification of niimerical 
operations and a more extended application of them. Among 
the Italians are evidences of an early maturity of arithmetic. 
Peacock ° says : The Tuscans generally, and the Florentines 
in particular, whose city was the cradle of the literature and 
arts of the thirteenth and fourteenth centuries, were celebrated 
for their knowledge of arithmetic and book-keeping, which 
were so necessary for their extensive commerce ; the Italians 
were in familiar possession of commercial arithmetic long 
before the other nations of Europe ; to them we are indebted 
for the formal introduction into books of arithmetic, under 
distinct heads, of questions in the single and double rule of 
three, loss and gain, fellowship, exchange, simple and com- 
pound interest, discount, and so on. 

There was also a slow improvement in the algebraic nota- 
tion. The Hindoo algebra possessed a tolerable symbolic 
notation, which was, however, completely ignored by the Mo- 
hammedans. In this respect, Arabic algebra approached 
much more closely to that of Diophantus, which can scarcely 



184 A HISTORY OF MATHEMATICS. 

be said to employ symbols in a systematic way. Leonardo of 
Pisa possessed no algebraic symbolism. Like the Arabs, he 
expressed the relations of magnitudes to each other by lines 
or in words. But in the mathematical writings of the monk 
Luca Pacioli (also called Lucas de Burgo sepulchri) symbols 
began to appear. They consisted merely in abbreviations of 
Italian words, such as p for piu (more), m for meno (less), co 
for cosa (the thing or unknown quantity). "Our present 
notation has arisen by almost insensible degrees as conven- 
ience suggested different marks of abbreviation to different 
authors ; and that perfect symbolic language which addresses 
itself solely to the eye, and enables us to take in at a glance 
the most complicated relations of quantity, is the result of a 
small series of small improvements." ** 

We shall now mention a few authors who lived during the 
thirteenth and fourteenth and the first half of the fifteenth 
centuries. About the time of Leonardo of Pisa (1200 a.d.), 
lived the Grerman monk Jordanus Nemorarius, who wrote a once 
famous work on the properties of numbers (1496), modelled 
after the arithmetic of Boethius. The most trifling numeral 
properties are treated with nauseating pedantry and prolixity. 
A practical arithmetic based on the Hindoo notation was 
also written by him. John Halifax (Sacro Bosco, died 1256) 
taught in Paris and made an extract from the Almagest con- 
taining only the most elementary parts of that work. This 
extract was for nearly 400 years a work of great popularity 
and standard authority. Other prominent writers are Albertus 
Magnus and George Purbach in Germany, and Roger Bacon in 
England. It appears that here and there some of our modern 
ideas were anticipated by writers of the Middle Ages. Thus, 
Nicole Oresme, a bishop in Normandy (died 1382), first con- 
ceived a notation of fractional powers, afterwards re-dis- 
covered by Stevinus, and gave rules for operating with them. 



EUBOPB DURING THE MIDDLB AOB8. 185 

His notation was totally different from ours. Thomas Brad- 
wardine, archbishop of Canterbury, studied star-polygons, — a 
subject which has recently received renewed attention. The 
first appearance of such polygons was with Pythagoras and 
his school. We next meet with such polygons in the geom- 
etry of Boethius and also in the translation of Euclid from 
the Arabic by Athelard of Bath. Bradwardine's philosophic 
writings contain discussions on the infinite and the infini- 
tesimal — subjects never since lost sight of. To England 
falls the honour of having produced the earliest European 
writers on trigonometry. The writings of Bradwardine, of 
Richard of Wallingford, and John Maudith, both professors 
at Oxford, and of Simon Bredon of Winchecombe, contain 
trigonometry drawn from Arabic sources. 

The works of the Greek monk Maximus Planudes, who lived 
in the first half of the fourteenth century, are of interest only 
as showing that the Hindoo numerals were then known in 
Greece. A writer belonging, like Planudes, to the Byzantine 
school, was Moschopulus, who lived in Constantinople in the 
early part of the fifteenth century. To him appears to be 
due the introduction into Europe of magic squares. He wrote 
a treatise on this subject. Magic squares were known to the 
Arabs, and perhaps to the Hindoos. Mediaeval astrologers 
and physicians believed them to possess mystical properties 
and to be a charm against plague, when engraved on silver 
plate. 

In 1494 was printed the Summa de Arithmetica, Geometria, 
Proportione et Propoi'tionalita, written by the Tuscan monk 
Lucas Padoli, who, as we remarked, first introduced symbols 
in algebra. This contains all the knowledge of his day on 
arithmetic, algebra, and trigonometry, and is the first com- 
prehensive work which appeared after the Liher Ahad of 
FibonaccL It contains little of importance which cannot be 



186 A HISTOBY OF MATHEMATICS. 

found in Fibonacci's great work, published three centuries 
earlier.* 

Perhaps the greatest result of the influx of Arabic learn- 
ing was the establishment of universities. What was their 
attitude toward mathematics ? The University of Paris, so 
famous at the beginning of the twelfth century under the 
teachings of Abelard^ paid but little attention to this science 
during the Middle Ages. Geometry was neglected, and Aris- 
totle's logic was the favourite study. In 1336, a rule was 
introduced that no student should take a degree without 
attending lectures on mathematics^ and from a commentary 
on the first six books of Euclid, dated 1536, it appears that 
candidates for the degree of A.M. had to give an oath that* 
they had attended lectures on these books.^ Examinations, 
when held at all, probably did not extend beyond the first 
book, as is shown by the nickname ^'magister matheseos," 
applied to the Theorem of Pjrthagoras, the last in the first 
book. More attention was paid to mathematics at the Univer- 
sUy of Prague, founded 1384. For the Baccalaureate degree, 
students were required to take lectures on Sacro Bosco's 
famous work on astronomy. Of candidates for the A.M. were 
required not only the six books of Euclid, but an additional 
knowledge of applied mathematics. Lectures were given on 
the Almagest. At the University of Leipzig, the daughter of 
Prague, and at Cologne, less work was required, and, as late 
as the sixteenth century, the same requirements were made at 
these as at Prague in the fourteenth. The universities of 
Bologna, Padua, Pisa, occupied similar positions to the ones 
in Germany, only that purely astrological lectures were given 
in place of lectures on the Almagest, At Oxford, in the 
middle of the fifteenth century, the first two books of Euclid 
were read.* 

Thus it will be seen that the study of mathematics was 



EUROPB DURING THB MIDDLE AGES. 137 

maintained at the universities only in a half-hearted manner. 
Ko great mathematician and teacher appeared, to inspire the 
students. The best energies of the schoolmen were expended 
upon the stupid subtleties of their philosophy. The genius 
of Leonardo of Pisa left no permanent impress upon the age, 
and another Renaissance of mathematics was wanted. 



MODERN EUROPE. 



We find it convenieDt to choose the time of the capture of 
Constantinople by the Turks as the date at which the Middle 
Ages ended and Modern Times began. In 1453^ the Turks 
battered the walls of this celebrated metropolis with cannon, 
and finally captured the city ; the Byzantine Empire fell, to 
rise no more. Calamitous as was this event to the East, it 
acted favourably upon the progress of learning in the West. 
A great number of learned Greeks fled into Italy, bringing 
with them precious manuscripts of Greek literature. This 
contributed vastly to the reviving of classic learning. Up 
to this time, Greek masters were known only through the 
often very corrupt Arabic manuscripts, but now they began 
to be studied from original sources and in their own language. 
The first English translation of Euclid was made in 1570 from 
the Greek by Sir Henry BiUingsley, assisted by John Dee,^ 
About the middle of the fifteenth century, printing was in- 
vented ; books became cheap and plentiful ; the printing-press 
transformed Europe into an audience-room. Near the close of 
the fifteenth century, America was discovered, and, soon after, 
the earth was circumnavigated. The pulse and pace of the 
world began to quicken. Men's minds became less servile; 
they became clearer and stronger. The indistinctness of 
thought, which was the characteristic feature of mediaeval 
learning, began to be remedied chiefly by the steady cultiva- 

138 



THE BENAI8SAKCB. 189 

tion of Pure Mathematics and Astronomy. Dogmatism was 
attacked; there arose a long struggle with the authority of 
the Church and the established schools of philosophy. The 
Copemican System was set up in opposition to the time-hon- 
oured Ptolemaic System. The long and eager contest between 
the two culminated in a crisis at the time of Galileo^ and 
resulted in the victory of the new system. Thus, by slow 
degrees, the minds of men were cut adrift from their old 
scholastic moorings and sent forth on the wide sea of scientific 
inquiiy, to discover new islands and continents of truth. 



THE RENAISSANCE. 

With the sixteenth century began a period of increased 
intellectual activity. The human mind made a vast effort to 
achieve its freedom. Attempts at its emancipation from 
Church authority had been made before, but they were stifled 
and rendered abortive. The first great and successful revolt 
against ecclesiastical authority was made in Germany. The 
new desire for judging freely and independently in matters 
of religion was preceded and accompanied by a growing spirit 
of scientific inquiry. Thus it was that, for a time, Germany 
led the van in science. She produced Eegiomontantia, Coper- 
nicus, Rhoeticus, Kepler, and Tycho Brake, at a period when 
France and England had, as yet, brought forth hardly any 
great scientific thinkers. This remarkable scientific produc- 
tiveness was no doubt due, to a great extent, to the commer- 
cial prosperity of Germany. Material prosperity is an essential 
condition for the progress of knowledge. As long as every 
individual is obliged to collect the necessaries for his subsist- 
ence, there can be no leisure for higher pursuits. At this 
time, Germany had accumulated considerable wealth. The 



140 A HISTORY OF MATHEMATICS. 

Hanseatic League commanded the trade of the North. Close 
commercial relations existed between Germany and Italy. 
Italy, too, excelled in commercial activity and enterprise. 
We need only mention Venice, whose glory began with the cru- 
sades, and Florence, with her bankers and her manufacturers 
of silk and wool. These two cities became great intellectual 
centres. Thus, Italy, too, produced men in art, literature, and 
science, who shone forth in fullest splendour. In ^t, Italy 
was the fatherland of what is termed the Kenaissance. 

For the first great contributions to the mathematical sciences 
we must, therefore, look to Italy and Grermany. In Italy 
brilliant accessions were made to algebra, in Germany to 
astronomy and trigonometry. 

On the threshold of this new era we meet in Germany with 
the figure of John Mueller, more generally called Regiomon- 
tanus (1436-1476). Chiefly to him we owe the revival of 
trigonometry. He studied astronomy and trigonometry at 
Vienna under the celebrated Gteorge Purbach. The latter 
perceived that the existing Latin translations of the Almagest 
were full of errors, and that Arabic authors had not remained 
true to the Greek original. Purbach therefore began to make 
a translation directly from the Greek. But he did not live to 
finish it. His work was continued by Regiomontanus, who 
went beyond his master. Regiomontanus learned the Greek 
language from Cardinal Bessarion, whom he followed to Italy, 
where he remained eight years collecting manuscripts from 
Greeks who had fled thither from the Turks. In addition to 
the translation of and the commentary on the Almagest, he 
prepared translations of the Conies of ApoUonius, of Archi- 
medes, and of the mechanical works of Heron. Regiomontanus 
and Purbach adopted the Hindoo sine in place of the Greek 
chord of double the arc. The Greeks and afterwards the Arabs 
divided the radius into 60 equal parts, and each of these again 



THE RENAISSANCE. 141 

into 60 smaller ones. The Hindoos expressed the length of 
the radius by parts of the circumference, saying that of the 
21^600 equal divisions of the latter, it took 3438 to measure 
the radius. Kegiomontanus, to secure greater precision, con- 
structed one table of sines on a radius divided into 600,000 
parts, and another on a radius divided decimally into 10,000,000 
divisions. He emphasised the use of the tangent in trigonom- 
etry. Following out some ideas of his master, he calculated 
a table of tangents. German mathematicians were not the 
first Europeans to use this function. In England it was known 
a century earlier to Bradwardine, who speaks of tangent {umbra 
recta) and cotangent {umbra versa), and to John Maudith. 
Begiomontanus was the author of an arithmetic and also of 
a complete treatise on trigonometry, containing solutions of 
both plane and spherical triangles. The form which he gave 
to trigonometry has been retained, in its main features, to the 
present day. 

Begiomontanus ranks among the greatest men that Germany 
has ever produced. His complete mastery of astronomy and 
mathematics, and his enthusiasm for them, were of far- 
reaching influence throughout Grermany. So great was his 
reputation, that Pope Sixtus IV. called him to Italy to 
improve the calendar. Begiomontanus left his beloved city 
of Nfimberg for Rome, where he died in the following year. 

After the time of Purbach and Begiomontanus, trigonome- 
try and especially the calculation of tables continued to occupy 
German scholars. More refined astronomical instruments were 
made, which gave observations of greater precision ; but these 
would have been useless without trigonometrical tables of cor- 
responding accuracy. Of the several tables calculated, that 
by Oeorg Joachim of Peldkirch in Tyrol, generally called 
Rlueticos, deserves special mention. He calculated a table of 
sines with the radius = 10,000,000,000 and from 10" to 10"; 



142 A HISTORY OF MATHEMATICS. 

and, later on, another with the radius = 1,000,000,000,000,000, 
and proceeding from 10" to 10". He began also the con- 
struction of tables of tangents and secants, to be carried to 
the same degree of accuracy ; but he died before finishing them. 
For twelve years he had had in continual employment several 
calculators. The work was completed by his pupil, Valentine 
OthO; in 1596. This was indeed a gigantic work, — a monu- 
ment of German diligence and indefatigable perseverance. 
The tables were republished in 1613 by Pitiscus, who spared 
no pains to free them of errors. Astronomical tables of 
so great a degree of accuracy had never been dreamed of 
by the Greeks, Hindoos, or Arabs. That Ehseticus was not a 
ready calculator only, is indicated by his views on trignomet- 
rical lines. Up to his time, the trigonometric functions had 
been considered always with relation to the arc ; he was the 
first to construct the right triangle and to make them depend 
directly upon its angles. It was from the right triangle that 
EhsBticus go this idea of calculating the hypotenuse ; i.e. he 
was the first to plan a table of secants. Good work in trigo- 
nometry was done also by Vieta and Romanus. 

We shall now leave the subject of trigonometry to witness 
the progress in the solution of algebraical equations. To do 
so, we must quit Germany for Italy. The first comprehensive 
algebra printed was that of Lucas Pacioli. He closes his 
book by saying that the solution of the equations a^ -f vix = n, 
a^-{-n = mx is as impossible at the present state of science as 
the quadrature of the circle. This remark doubtless stimu- 
lated thought. The first step in the algebraic solution of 
cubics was taken by Scipio Ferro (died 1526), a professor of 
mathematics at Bologna, who solved the equation x^ -f mx = n. 
Nothing more is known of his discovery than that he imparted 
it to his pupil, Floridas, in 1505. It was the practice in those 
days and for two centuries afterwards to keep discoveries 



THE BENAISSAKCE. 148 

secret, in order to secure by that means an advantage over 
rivals by proposing problems beyond their reach. This prac- 
tice gave rise to numberless disputes regarding the priority of 
inventions. A second solution of cubics was given by Nicolo 
of Brescia (1506(?)-1657). When a boy of six, Nicolo was 
so badly cut by a French soldier that he never again gained 
tl^e free use of his tongue. Hence he was called Tartaglia, 
i.e. the stammerer. His widowed mother being too poor to 
pay his tuition in school, he learned to read and picked up a 
knowledge of Latin, Greek, and mathematics by himself. 
Possessing a mind of extraordinary power, he was able to 
appear as teacher of mathematics at an early age. In 1530, 
one Colla proposed him several problems, one leading to the 
equation a^+pa? = q. Tartaglia found an imperfect method 
for solving this, but kept it secret. He spoke about his secret 
in public and thus caused Ferro's pupil, Floridas, to proclaim 
his own knowledge of the form sc* -f mx = n. Tartaglia, believ- 
ing him to be a mediocrist and braggart, challenged him to a 
public discussion, to take place on the 22d of February, 1535. 
Hearing, meanwhile, that his rival had gotten the method 
from a deceased master, and fearing that he would be beaten 
in the contest, Tartaglia put in all the zeal, industry, and 
skill to find the rule for the equations, and he succeeded in it 
ten days before the appointed date, as he himself modestly 
says.' The most difficult step was, no doubt, the passing from 
quadratic irrationals, used in operating from time of old, to 
cubic irrationals. Placing aj=:^ — Vw, Tartaglia perceived 
that the irrationals disappeared from the equation a?+ mx=:n, 
making n = < — w. But this last equality, together with 
(Jm)* = ^tt, gives at once 



-<(DHlH "=V(i)'HlM 



144 A HISTORY OF MATHEMATICS. 

This is Tartaglia's solution oi a^ + mxzsn. On the 13th of 
February, he found a similar solution for a? = mx + n. The 
contest began on the 22d. Each contestant proposed thirty 
problems. The one who could solve the greatest number within 
fifty days should be the victor. Jartaglia solved the thirty 
problems proposed by Floridas in two hours ; Floridas could 
not solve any of Tartaglia's. From now on, Tartaglia studied 
cubic equations with a will. In 1541 he discovered a general 
solution for the cubic a? ±pQi?^ ± q, by transforming it into 
the form a^ ± mx = ± n. The news of Tartaglia's victory 
spread all over Italy. Tartaglia was entreated to make known 
his method, but he declined to do so, saying that after his 
completion of the translation from the Greek of Euclid and 
Archimedes, he would publish a large algebra containing his 
method. But a scholar from Milan, named Hieronimo Cardano 
(1601-1676), after many solicitations, and after giving the 
most solemn and sacred promises of secrecy, succeeded in 
obtaining from Tartaglia a knowledge of his rules. 

At this time Cardan was writing his Are Magna, and he 
knew no better way to crown his work than by inserting the 
much sought for rules for solving cubics. Thus Cardan broke 
his most solemn vows, and published in 1646 in his Ars Magna 
Tartaglia's solution of cubics. Tartaglia became desperate. 
His most cherished hope, of giving to the world an immortal 
work which should be the monument of his deep learning and 
power for original research, was suddenly destroyed ; for the 
crown intended for his work had been snatched away. His 
first step was to write a history of his invention ; but, to com- 
pletely annihilate his enemies, he challenged Cardan and his 
pupil Lodovico Ferrari to a contest: each party should propose 
thirty-one questions to be solved by the other within fifteen 
days. Tartaglia solved most questions in seven days, but the 
other party did not send in their solution before the expiration 



THE BENAISSANCB. 145 

of the fifth month; moreover, all their solutions except one 
were wrong. A replication and a rejoinder followed. Endless 
were the problems proposed and solved on both sides. The 
dispute produced much chagrin and heart-burnings to the par- 
ties, and to Tartaglia especially, who met with many other 
disappointments. After having recovered himself again, Tar- 
taglia began, in 1556, the publication of the work which he 
had had in his mind for so long; but he died before he reached 
the consideration of cubic equations. Thus the fondest wish 
of his life remained unfulhlled ; the man to whom we owe the 
greatest contribution to algebra made in the sixteenth century 
was forgotten, and his method came to be regarded as the dis- 
covery of Cardan and to be called Cardan's solution. 

Bemarkable is the great interest that the solution of cubics 
excited throughout Italy. It is but natural that after this 
great conquest mathematicians should attack bi-quadratic equa- 
tions. As in the case of cubics, so here, the first impulse was 
given by Colla, who, in 1540, proposed for solution the equa- 
tion a^ -f- 6 aj* -h 36 = 60 a;. To be sure. Cardan had studied 
particular cases as early as 1539. Thus he solved the equation 
13«* = a?*-f-2a5'-f-2a;-f-l by a process similar to that em- 
ployed by Diophantus and the Hindoos; namely, by adding 
to both sides 3a^ and thereby rendering both numbers 
complete squares. But Cardan failed to find a general solu- 
tion; it remained for his pupil Ferrari to prop the reputa- 
tion of his master by the brilliant discovery of the general 
solution of bi-quadratic equations. Ferrari reduced Colla's 
equation to the form (a:*-f- 6)' = 60aj-f-6aj*. In order to 
give also the right member the form of a complete square 
he added to both members the expression 2(aj*-f- 6)y-|-y', 
containing a new unknown quantity y. This gave him (aj* -f- 6 
+ y)«=(6 + 2y)aj*-f-60a;+(12y-f-y"). The condition that 
the right member be a complete square is expressed by the 



146 A HISTORY OF MATHBMATIGS. 

cubic equation (2 y 4- 6) (12 y -|- y*) = 900. Extracting the 

square root of the bi-quadratic, he got aj* + 6 + y = 05 'V2y~+^ 

900 
4- Solving the cubic for y and substituting, it re- 

v2y-|-6 

mained only to determine x from the resulting quadratic. 
Ferrari pursued a similar method with other numerical bi- 
quadratic equations.^ Cardan had the pleasure of publishing 
this discovery in his Ars Magna in 1545. Ferrari's solution 
is sometimes ascribed to BombeUi, but he is no more the dis- 
coverer of it than Cardan is of the solution called by his 
name. 

To Cardan algebra is much indebted. In his Are Magna 
he takes notice of negative roots of an equation, calling them 
fictitiouSj while the positive roots are called real. Imaginary 
roots he does not consider; cases where they appear he calls 
impossible. Cardan also observed the difficulty in the irre- 
ducible case in the cubics, which, like the quadrature of the 
circle, has since " so much tormented the perverse ingenuity of 
mathematicians." But he did not understand its nature. It re- 
mained for Raphael Bombelli of Bologna, who published in 1572 
an algebra of great merit, to point out the reality of the appar- 
ently imaginary expression which the root assumes, and thus 
to lay the foundation of a more intimate knowledge of imagi- 
nary quantities. 

After this brilliant success in solving equations of the third 
and fourth degrees, there was probably no one who doubted, 
that with aid of irrationals of higher degrees, the solution of 
equations of any degree whatever could be found. But all 
attempts at the algebraic solution of the quintic were fruitless, 
and, finally, Abel demonstrated that all hopes of finding alge- 
braic solutions to equations of higher than the fourth degree 
were purely Utopian. 

Since no solution by radicals of equations of higher degrees 



THB BBKAISSANGB. 147 

could be found, there remained nothing else to be done than 
tike devising of rules by which at least the numerical YaJues of 
^he roots could be ascertained. Cardan applied the Hindoo 
rule of " false position " (called by him regula aurea) to the 
cubic, but this mode of approximating was exceedingly rough. 
An incomparably better method was invented by Franciscus 
Yieta, a French mathematician, whose transcendent genius 
enriched mathematics with several important innovations. 
Taking the equation f{x) = Q, wherein f(x) is a polynomial 
containing different powers of x, with numerical coefficients, 
and Q is a given number, Vieta first substitutes in f(x) a 
known approximate value of the root, and then shows that 
another figure of the root can be obtained by division. A repe- 
tition of the same process gives the next figure of the root, 
and so on. Thus, in aj*-f 14 a; = 7929, taking 80 for the approx- 
imate root, and placing a; = 80 -f 6, we get 

(80 + 6)« 4- 14 (80 4- &) = 7929, 
Or 1746 + &' = 409. 

Since 1746 is much greater than 6', we place 1746 = 409, and 
obtain thereby 6 = 2. Hence the second approximation is 82. 
I^ut a? = 82 4- c, then (82 4- c)«4- 14(82 4- c) = 7929, or 178c4-c* 
a= 57. As before, place 178c= 67, then c = .3, and the third 
^approximation gives 82.3. Assuming x = 82.3 4- d, and substi- 
tuting, gives 178.6 d 4- <P = 3.51, and 178.6 d = 3.51, .-. d = .01 ; 
giving for the fourth approximation 82.31. In the same way, 
e = .009, and the value for the root of the given equation is 
82.319... For this process, Vieta was greatly admired by his 
contemporaries. It was employed by Harriot, Oughtred, Pell, 
and others. Its principle is identical with the main principle 
involved in the methods of approximation of Newton and 
Homer. The only change lies in the arrangement of the 



148 A UlSTOBY OF MATHEMATICS. 

woik. TMb alteration vaa made to afford £aciUtf aod gecar- 
ity iu the process of evolatiou of the root 

We pause a momeat to sketch the life of Vieta, the most 
eminent French mathematician of the sixteenth centnijr. He 
was bom in Foitou in lo40, and died in 1603 at Paris. He 
was employed throughout life in the service of the state, 
under Henry IIL and Henry IV. He was, therefore, not a 
mathematician by profession, but his love for the science was 
so great that he remained in his chamber studying, sometimes 
several days in succession, withoat eating and sleepii^ more 
than was necessary to sustain himself. So great devotion to 
abstract science is the more remarkable, because he lived at a 
time of incessant political and religious turmoil. During the 
war against Spain, Vieta rendered service to Heniy IV. by 
deciphering intercepted letters written in a species of cipher, 
and addressed by the Spanish Court to their governor of Neth- 
erlands. The Spaniards attributed the discovery of the key 
to mngic. 

An embassador from Netherlands once told Henry IV. that 
Frani-o did not possess a single geometer capable of solving a 
pntblom propounded to geometers by a Belgian mathematician, 
Adritwus Bomanus. It was the solution of the equation <rf 
the forty -fifth degree : — 



Hcnrt- IV. oaUed Vieta, who^ hanng alnadj p 

iiiTesti)n>tions, saw at once that this awe-inspiring J 

wiH simply the ci^aation by which C=2sin* * 

ill terms of j = 2sin(^*; that, siiio*- 4r.= :5-.S-5, 

ut>cea$ary only to divide an angle onoe inta Ti ^q\ial parU, and 

Uion twioe intnA — "'ti"""" «bifhooold be effected byoor^ 

n>S[>o«tiiu8 etiiui : * u-.d tWn! degrw* 

was tlte di«contr> i r..<iU tDtbtsequatioOiUSt' 



THE BEKAISSANCE. 149 

of only one. The reason why he did not find 45 solutions, is 
that the remaining ones involve negative sines, which were 
unintelligible to him. Detailed investigations on the famous 
old problem of the section of an angle into an odd number of 
equal parts, led Vieta to the discovery of a trigonometrical 
solution of Cardan's irreducible case in cubics. He applied 
the equation (2cos^<^)'— 3(2cos^<^) = 2cos<^ to the solution 
of as* — 3a*a5 = a'6, when a>^6, by placing a;=:2acos^<^, and 
determining <^ from 6 = 2 a cos <^. 

The main principle employed by him in the solution of 
equations is that of reduction. He solves the quadratic by 
making a suitable substitution which will remove the term 
containing x to the first degree. Like Cardan, he reduces the 
general expression of the cubic to the form a^+ma;+n=0; 
then, assuming a; = (^ a — z') -s- 2 and substituting, he gets 
a^ — te*— *^a' = 0. Putting r* = y, he has a quadratic. In 
the solution of bi-quadratics, Vieta still remains true to his 
principle of reduction. This gives him the well-known cubic 
resolvent. He thus adheres throughout to his favourite prin- 
ciple, and thereby introduces into algebra a uniformity of 
method which claims our lively admiration. In Vieta's algebra 
we discover a partial knowledge of the relations existing 
between the coefficients and the roots of an equation. He 
shows that if the coefficient of the second term in an equation 
of the second degree is minus the sum of two numbers whose 
product is the third term, then the two numbers are roots of 
the equation. Vieta rejected all except positive roots ; hence 
it was impossible for him to fully perceive the relations in 
question. 

The most epoch-making innovation in algebra due to Vieta 
is the denoting of general or indefinite quantities by letters 
of the alphabet. To be sure, Regiomontanus and Stifel in 
Grermany, and Cardan in Italy, used letters before him, but 



150 A HISTORY OF MATHEBiATICS. 

Vieta extended the idea and first made it an essential part of 
algebra. The new algebra was called by him logistica speciosa 
in distinction to the old logistica numeroaa. Vieta's formalism 
differed considerably from that of to-day. The equation a* + 
Sa*b + SaV + V = {a + by was written by him "a cubus -f- b 
in a quadr. 3 + a in b quadr. 3 + 6 cubo sequalia a'+~5 cubo.'' 
In numerical equations the unknown quantity was denoted by 
Ny its square by Q, and its cube by C Thus the equation 
aj8-8a*4-16aj = 40 was written 1 C - S Q + 16 N cequal, 40. 
Observe that exponents and our symbol ( = ) for equality were 
not yet in use; but that Vieta employed the Maltese cross 
(+) as the short-hand symbol for addition, and the ( — ) for 
subtraction. These two characters had not been in general 
use before the time of Vieta. " It is very singular," says 
Hallam, "that discoveries of the greatest convenience, and, 
apparently, not above the ingenuity of a village schcJolmaster, 
should have been overlooked by men of extraordinary acute- 
ness like Tartaglia, Cardan, and Ferrari; and hardly less so 
that, by dint of that acuteness, they dispensed with the aid of 
these contrivances in which we suppose that so much of the 
utility of algebraic expression consists." Even after improve- 
ments in notation were once proposed, it was with extreme 
slowness that they were admitted into general use. They 
were made oftener by accident than design, and their authors 
had little notion of the effect of the change which they were 
making. The introduction of the -f and — symbols seems 
to be due to the Germans, who, although they did not enrich 
algebra during the Renaissance with great inventions, as did 
the Italians, still cultivated it with great zeal. The arith- 
metic of John Widmann, printed a.d. 1489 in Leipzig, is the 
earliest book in which the -f- and — symbols have been found. 
There are indications leading us to surmise that they were in 
use first among merchants. They occur again in the arith- 



THE RENAISSANCE. 151 

metic of Grammateus, a teacher at the University of Vienna. 

His pupil, Christoff Rudolff, the writer of the first text-book 

on algebra in the Grerman language (printed in 1525), employs 

these symbols also. So did Stifel, who brought out a second 

edition of Eudolff's Coss in 1553. Thus, by slow degrees, 

their adoption became universal. There is another short-hand 

symbol of which we owe the origin to the (Jermans. In a 

manuscript published sometime in the fifteenth century, a dot 

placed before a number is made to signify the extraction of a 

root of that number. This dot is the embryo of our present 

symbol for the square root. Christoff Eudolff, in his algebra, 

remarks that " the radix quadrata is, for brevity, designated 

in his algorithm with the character y/, as y^^." Here the dot 

has grown into a symbol much like our own. This same 

Symbol was used by Michael Stifel Our sign of equality is due 

to Robert Recorde (1510-1558), the author of The Whetstone of 

Witte (1557), which is the first English treatise on algebra. 

fie selected this symbol because no two things could be more 

^ual than two parallel lines =. The sign -^ for division was 

&st used by Johann Heinrich Eahn, a Swiss, in 1659, and 

uras introduced in England by John Pell in 1668. 

Michael Stifel (1486?-1567), the greatest German algebraist 
of the sixteenth century, was born in Esslingen, and died in 
Jena. He was educated in the monastery of his native place, 
and afterwards became Protestant minister. The study of the 
significance of mystic numbers in Revelation and in Daniel 
drew him to mathematics. He studied German and Italian 
works, and published in 1544, in Latin, a book entitled 
Arithmetica Integra, Melanchthon wrote a preface to it. Its 
three parts treat respectively of rational numbers, irrational 
numbers, and algebra. Stifel gives a table containing the nu- 
merical values of the binomial coefficients for powers below the 
18th. He observes an advantage in letting a geometric progres- 



* t 



152 A HISTORY OP MATHEMATICS. 

sion correspond to an arithmetical progression, and arriyes at 
the designation of integral powers by numbers. Here are the 
germs of the theory of exponents. In 1545 Stifel published 
an arithmetic in German. His edition of Rudolff's Cosa con- 
tains rules for solving cubic equations, derived from the 
the writings of Cardan. 

We remarked above that Vieta discarded negative roots of 
equations. Indeed, we find few algebraists before and during 
the Eenaissance who understood the significance even of 
negative quantities. Fibonacci seldom uses them. Pacioli 
states the rule that "minus times minus gives plus," but 
applies it really only to the development of the product of 
(a — 6) (c — d) ; purely negative quantities do not appear in 
his work. The great German "Cossist" (algebraist), Michael 
Stifel, speaks as early as 1544 of numbers which are " absurd '' 
or " fictitious below zero," and which arise when " real numbers 
above zero " are subtracted from zero. Cardan, at last, speaks 
of a " pure minus " ; " but these ideas," says Hankel, "remained 
sparsely, and until the beginning of the seventeenth century, 
mathematicians dealt exclusively with absolute positive quan- 
tities." The first algebraist who occasionally places a purely 
negative quantity by itself on one side of an equation, is 
Harriot in England. As regards the recognition of negative 
roots. Cardan and Bombelli were far in advance of all writers 
of the Renaissance, including Vieta. Yet even they mentioned 
these so-called false or fictitious roots only in passing, and 
without grasping their real significance and importance. On 
this subject Cardan and Bombelli had advanced to about the 
same point as had the Hindoo Bhaskara, who saw negative roots, 
but did not approve of them. The generalisation of the con- 
ception of quantity so as to include the negative, was an 
exceedingly slow and diflScult process in the development of 
algebra. 



THE BBNAISSANCB. 158 

We shall now consider the history of geometry during the 
Renaissance. Unlike algebra, it made hardly any progress. 
The greatest gain was a more intimate knowledge of Greek 
geometry. No essential progress was made before the time of 
Descartes. Begiomontanus, Xylander of Augsburg, Tartaglia, 
Commandinus of Urbino in Italy, Maurolycus, and others, 
made translations of geometrical works from the Greek. John 
Werner of Kiirnberg published in 1522 the first work on 
conies which appeared in Christian Europe. Unlike the 
geometers of old, he studied the sections in relation with the 
oone, and derived their properties directly from it. This mode 
of studying the conies was followed by BCaurolycus of Messina 
(1494-1575). The latter is, doubtless, the greatest geometer 
of the sixteenth century. From the notes of Pappus, he 
attempted to restore the missing fifth book of ApoUonius 
on maxima and minima. His chief work is his masterly 
and original treatment of the conic sections, wherein he dis- 
cusses tangents and asymptotes more fully than ApoUonius 
had done, and applies them to various physical and astronomi- 
cal problems. 

The foremost geometrician of Portugal was Nonius; of 
France, before Vieta, was Peter Ramus, who perished in the 
massacre of St. Bartholomew. Vieta possessed great famil- 
iarity with ancient geometry. The new form which he gave 
to algebra, by representing general quantities by letters, en- 
abled him to point out more easily how the construction of 
the roots of cubics depended upon the celebrated ancient prob- 
lems of the duplication of the cube and the trisection of an 
angle. He reached the interesting conclusion that the former 
problem includes the solutions of all cubics in which the radi- 
cal in Tartaglia's formula is real, but that the latter problem 
includes only those leading to the irreducible case. 

The problem of the quadrature of the circle was revived in 



154 A HISTORY 02* MATHEMATIOS. 

this age, and was zealously studied even by men of eminence 
and mathematical ability. The army of circle-squarers became 
most formidable during the seventeenth century. Among the 
first to revive this problem was the German Cardinal Nicolaus 
Cttsanus (died 1464) , who had the reputation of being a great 
logician. His fallacies were exposed to full view by Begio- 
montanus. As in this case, so in others, every quadrator of 
note raised up an opposing mathematician : Orontius was met 
by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus 
Romanus, and Clavius ; A. Quercu by Peter Metius. Two 
mathematicians of Netherlands, Adrianus Romanus and Ludolph 
van Ceulen, occupied themselves with approximating to the 
ratio between the circumference and the diameter. The for- 
mer carried the value ir to 15, the latter to 35, places. The 
value of IT is therefore often named " Ludolph's number." His 
performance was considered so extraordinary, that the num- 
bers were cut on his tomb-stone in St. Peter's church-yard, at 
Leyden. Komanus was the one who propounded for solution 
that equation of the forty-fifth degree solved by Vieta. On 
receiving Vieta's solution, he at once departed for Paris, to 
make his acquaintance with so great a master. Vieta pro- 
posed to him the Apollonian problem, to draw a circle touching 
three given circles. " Adrianus Romanus solved the problem 
by the intersection of two hyperbolas ; but this solution did not 
possess the rigour of the ancient geometry. Vieta caused him 
to see this, and then, in his turn, presented a solution which 
had all the rigour desirable."** Romanus did much toward 
simplifying spherical trigonometry by reducing, by means of 
certain projections, the 28 cases in triangles then considered 
to only six. 

Mention must here be made of the improvements of the 
Julian calendar. The yearly determination of the movable 
feasts had for a long time been connected with an untold 



THE EENAI8SANCB. 155 

amount of confusion. The rapid progress of astronomy led 
to the consideration of this subject, and many new calendars 
were proposed. Pope Gregory XIII. convoked a large number 
of mathematicians, astronomers, and prelates, who decided 
upon the adoption of the calendar proposed by the Jesuit 
lilins Clavitts. To rectify the errors of the Julian calendar 
it was agreed to write in the new calendar the 15th of 
October immediately after the 4th of October of the year 
1582. The Gregorian calendar met with a great deal of oppo- 
sition both among scientists and among Protestants. Clavius, 
who ranked high as a geometer, met the objections of the 
former most ably and effectively ; the prejudices of the latter 
passed away with time. 

The passion for the study of mystical properties of numbers 
descended from the ancients to the moderns. Much was 
written on numerical mysticism even by such eminent men 
as Pacioli and Stifel. The Nuinerorum Mysteria of Peter 
Bungus covered 700 quarto pages. He worked with great 
industry and satisfaction on 666, which is the number of the 
beast in Revelation (xiii. 18), the symbol of Antichrist. He 
reduced the name of the ' impious ' Martin Luther to a form 
which may express this formidable number. Placing a = 1, 
6 = 2, etc., A: = 10, 1= 20, etc., he finds, after misspelling the 
name, that M(30) A^jj R^ao, ^(voo) I(») ^(o -^(20) m«o) T(ioo) ^(S) R(») ^{d 
constitutes the number required. These attacks on the great 
reformer were not unprovoked, for his friend, Michael Stifel, 
the most acute and original of the early mathematicians of 
Germany, exercised an equal ingenuity in showing that the 
above number referred to Pope Leo X., — a demonstration 
which gave Stifel unspeakable comfort.** 

Astrology also was still a favourite study. It is well 
known that Cardan, Maurolycus, Regiomontanus, and many 
other eminent scientists who lived at a period even later than 



156 A HISTORY OF MATHEMATICS. 

this, engaged in deep astrological study ; but it is not so gen- 
erally known that besides the occult sciences already named, 
men engaged in the mystic study of star-polygons and magic 
squares. " The pentagramma gives you pain," says Faust to 
Mephistopheles. It is of deep psychological interest to see 
scientists, like the great Kepler, demonstrate on one page a 
theorem on star-polygons, with strict geometric rigour, while 
on the next page, perhaps, he explains their use as amulets 
or in conjurations.^ Playfair, speaking of Cardan as an astrol- 
oger, calls him ''a melancholy proof that there is no folly 
or weakness too great to be united to high intellectual attain- 
ments."" Let our judgment not be too harsh. The period 
under consideration is too near the Middle Ages to admit of 
complete emancipation from mysticism even among scientists. 
Scholars like Kepler, Napier, Albrecht Duerer, while in the 
van of progress and planting one foot upon the firm ground 
of truly scientific inquiry, were still resting with the other 
foot upon the scholastic ideas of preceding ages. 



VIETA TO DESCARTES. 

The ecclesiastical power, which in the ignorant ages was an 
unmixed benefit, in more enlightened ages became a serious 
evil. Thus, in France, during the reigns preceding that of 
Henry IV., the theological spirit predominated. This is pain- 
fully shown by the massacres of Vassy and of St. Bartholo- 
mew. Being engaged in religious disputes, people had no 
leisure for science and for secular literature. Hence, down 
to the time of Henry IV., the French " had not put forth a 
single work, the destruction of which would now be a loss to 
Europe." In England, on the other hand, no religious wars 
were waged. The people were comparatively indifferent about 



VIBTA TO DESGABTES. 167 

religious strifes ; thej concentrated their ability upon secular 
matters, and acquired, in the sixteenth century, a literature 
which is immortalised by the genius of Shakespeare and 
Spenser. This great literary age in England was followed 
by a great scientific age. At the close of the sixteenth cen- 
tuiy, the shackles of ecclesiastical authority were thrown off 
by France. The ascension of Henry IV. to the throne was 
followed in 1598 by the Edict of Nantes, granting freedom 
of worship to the Huguenots, and thereby terminating religious 
wars. The genius of the French nation now began to blossom. 
Cardinal Eichelieu, during the reign of Louis XIII., pursued 
the broad policy of not favouring the opinions of any sect, but 
of promoting the interests of the nation. His age was re- 
markable for the progress of knowledge. It produced that 
great secular literature, the counterpart of which was found 
in England in the sixteenth century. The seventeenth cen- 
tury was made illustrious also by the great French mathema- 
ticians, Roberval, Descartes, Desargues, Fermat, and Pascal. 

More gloomy is the picture in Grermany. The great changes 
which revolutionised the world in the sixteenth century, and 
which led England to national greatness, led Germany to 
degradation. The first effects of the Reformation there were 
salutary. At the close of the fifteenth and during the six- 
teenth century, Germany had been conspicuous for her scien- 
tific pursuits. She had been the leader in astronomy and 
trigonometry. Algebra also, excepting for the discoveries in 
cubic equations, was, before the time of Vieta, in a more 
advanced state there than elsewhere. But at the beginning 
of the seventeenth century, when the sun of science began to 
rise in France, it set in Germany. Theologic disputes and 
religious strife ensued. The Thirty Years' War (1618-1648) 
proved ruinous. The German empire was shattered, and 
became a mere lax confederation of petty despotisms. Com- 



158 A HISTOBY OF MATHEMATICS. 

merce was destroyed; national feeling died out. Art disap- 
peared^ and in literature there was only a slavish imitation 
of French artificiality. Nor did Germany recover from this 
low state for 200 years ; for in 1756 began another struggle, 
the Seven Years' War, which turned Prussia into a wasted 
land. Thus it followed that at the beginning of the seven- 
teenth century, the great Kepler was the only Grerman mathje- 
matician of eminence, and that in the interval of 200 years 
between Kepler and Grauss, there arose no great mathematician 
in Germany excepting Leibniz. 

Up to the seventeenth century, mathematics was cultivated 
but little in Great Britain. During the sixteenth century, she 
brought forth no mathematician comparable with Vieta, Stifel, 
or Tartaglia. But with the time of Recorde, the English 
became conspicuous for numerical skill. The first important 
arithmetical work of English authorship was published in 
Latin in 1622 by Cuthbert Tonstall (1474-1659). He had 
studied at Oxford, Cambridge, and Padua, and drew freely 
from the works of Pacioli and Regiomontanus. Reprints of 
his arithmetic appeared in England and France. After 
Recorde the higher branches of mathematics began to be 
studied. Later, Scotland brought forth Napier, the inventor 
of logarithms. The instantaneous appreciation of their value 
is doubtless the result of superiority in calculation. In Italy, 
and especially in France, geometry, which for a long time had 
been an almost stationary science, began to be studied with 
success. Galileo, Torricelli, Roberval, Fermat, Desargues, 
Pascal, Descartes, and the English Wallis are the great revo- 
lutioners of this science. Theoretical mechanics began to be 
studied. The foundations were laid by Fermat and Pascal 
for the theory of numbers and the theory of probability. 

We shall first consider the improvements made in the art 
of calculating. The nations of antiquity experimented thou- 



VIBTA TO DESCARTES. 159 

sands of years upon numeral notations before they happened 
to strike upon the so-called '^ Arabic notation." In the simple 
expedient of the cipher, which was introduced by the Hindoos 
about the fifth or sixth century after Christ, mathematics re- 
ceived one of the most powerful impulses. It would seem that 
after the " Arabic notation " was once thoroughly understood, 
decimal fractions would occur at once as an obvious extension 
of it. But ''it is curious to think how much science had 
attempted in physical research and how deeply numbers had 
been pondered, before it was perceived that the all-powerful 
simplicity of the 'Arabic notation' was as valuable and as 
manageable in an infinitely descending as in an infinitely 
ascending progression." " Simple as decimal fractions appear 
to us, the invention of them is not the result of one mind or 
even of one age. They came into use by almost imperceptible 
degrees. The first mathematicians identified with their his- 
tory did not perceive their true nature and importance, and 
failed to invent a suitable notation. The idea of decimal 
fractions makes its first appearance in methods for approxi- 
mating to the square roots of numbers. Thus John of Seville, 
presumably in imitation of Hindoo rules, adds 2n ciphers 
to the number, then finds the square root, and takes this 
as the numerator of a fraction whose denominator is 1 fol- 
lowed by n ciphers. The same method was followed by 
Cardan, but it failed to be generally adopted even by his 
Italian contemporaries ; for otherwise it would certainly have 
been at least mentioned by Cataldi (died 1626) in a work 
devoted exclusively to the extraction of roots. Cataldi finds 
the square root by means of continued fractions — a method 
ingenious and novel, but for practical purposes inferior to 
Cardan's. Orontius Finaeus (died 1555) in France, and Wil- 
liam Buckley (died about 1550) in England extracted the 
square root in the same way as Cardan and John of Seville. 



160 A HISTORY OF MATHEMATICS. 

The invention of decimals is frequently attributed to Regio- 
montanus, on the ground that instead of placing the sinus 
totus, in trigonometry, equal to a multiple of 60, like the 
Greeks, he put it = 100,000. But here the trigonometrical 
lines were expressed in integers, and not in fractions. Though 
he adopted a decimal division of the radius, he and his suc- 
cessors did not apply the idea outside of trigonometry and, 
indeed, had no notion whatever of decimal fractions. To 
Simon Stevin of Bruges in Belgium (1548-1620), a man who 
did a great deal of work in most diverse fields of science, we 
owe the first systematic treatment of decimal fractions. In 
his La Disme (1585) he describes in very express terms the 
advantages, not only of decimal fractions, but also of the 
decimal division in systems of weights and measures. Stevin 
applied the new fractions '^to all the operations of ordinary 
arithmetic." * What he lacked was a suitable notation. In 
place of our decimal point, he used a cipher ; to each place in 
the fraction was attached the corresponding index. Thus, in 

0128 

his notation, the number 5.912 would be 5912 or 5(o)9®l®2(D. 
These indices, though cumbrous in practice, are of interest, 
because they are the germ of an important innovation. To 
Stevin belongs the honour of inventing our present mode of 
designating powers and also of introducing fractional expo- 
nents into algebra. Strictly speaking, this had been done much 
earlier by Oresme, but it remained wholly unnoticed. Not even 
Stevin's innovations were immediately appreciated or at once 
accepted, but, unlike Oresme's, they remained a secure posses- 
sion. No improvement was made in the notation of decimals 
till the beginning of the seventeenth century. After Stevin, 
decimals were used by Joost Biirgi, a Swiss by birth, who pre- 
pared a manuscript on arithmetic soon after 1592, and by 
Johann Hartmann Beyer, who assumes the invention as his own. 
In 1603, he published ^d* Frankfurt on the Main a Logistica 



VIBTA TO DES0AliT£8. 161 

Dedmalis. With Burgi, a zero placed underneath the digit 
in unit's place answers as sign of separation. Beyer's nota- 
tion resembles Stevin's. The decimal point, says Peacock, is 
due to Napier, who in 1617 published his Babdologiay contain- 
ing a treatise on decimals, wherein the decimal point is used 
in one or two instances. In the English translation of Napier's 
Mirifid logarithmorum canonia description executed by Edward 
Wright in 1616, and corrected by the author, the decimal 
point occurs in the tables. There is no mention of decimals 
in English arithmetics between 1619 and 1631. Oughtred in 
1631 designates the fraction .56 thus, 0|56. Albert Cfirardj 
a pupil of Stevin, in 1629 uses the point on one occasion. 
John Wallis in 1657 writes 12 1 345 , but afterwards in his 
algebra adopts the usual point. De Morgan says that '^ to the 
first quarter of the eighteenth century we must refer not only 
the complete and final victory of the decimal point, but also 
that of the now universal method of performing the operations 
of division and extraction of the square root."*^ We have 
dwelt at some length on the progress of the decimal notation, 
because ** the history of language ... is of the highest order 
of interest, as well as utility: its suggestions are the best 
lesson for the future which a reflecting mind can have." ^ 

The miraculous powers of modern calculation are due to 
three inventions: the Arabic Notation, Decimal Fractions, and 
Logarithms. The invention of logarithms in the first quarter 
of the seventeenth century was admirably timed, for Kepler 
was then examining planetary orbits, and Galileo had just 
turned the telescope to the stars. During the Renaissance 
German mathematicians had constructed trigonometrical 
tables of great accuracy, but this greater precision enormously 
increased the work of the calculator. It is no exaggeration 
to say that the invention of logariJims ''by shortening the 
labours doubled the life of the astron^ ler." Logarithms were 



162 A HISTORY OF MATHBICATICS. 

invented by John Napier, Baron of Merchiston, in Scotland 
(1550-1617). It is one of the greatest curiosities of the 
history of science that Napier constructed logarithms before 
exponents were used. To be sure, Stifel and Stevin made 
some attempts to denote powers by indices, but this notation 
was not generally known, — not even to Han'iotj whose algebra 
appeared long after Napier's death. That logarithms flow 
naturally from the exponential symbol was not observed until 
much later. It was Euler who first considered logarithms as 
being indices of powers. What, then, was Napier's line of 
thought ? 

Let AB be a definite line, DE a line extending from D in- 
definitely. Imagine two points starting at the same moment ; 

the one moving from 
-^ ^ ^ A toward B, the other 

from D toward E, Let 
j5 f e the velocity during the 

first moment be the 
same for both : let that of the point on line DE be uniform ; 
but the velocity of the point on AB decreasing in such a way 
that when it arrives at any point C, its velocity is proportional 
to the remaining distance BC, While the first point moves 
over a distance -4C, the second one moves over a distance DF. 
Napier calls DF the logarithm of BC, 

Napier's process is so unique and so different from all other 
modes of presenting the subject that there cannot be the 
shadow of a doubt that this invention is entirely his own ; it 
is the result of unaided, isolated speculation. He first sought 
the logarithms only of sines ; the line AB was the sine of 90** 
and was taken = 10^ ; BC was the sine of the arc, and DF its 
logarithm. We notice that as the motion proceeds, BC de- 
creases in geometrical progression, while DF increjises in 
arithmetical progression. Let AB = a = 10^, let x = DFy 



VIETA TO DESCARTES. 168 

yssBCj then AC = a — y. The velocity of the point C is 

V^ — y) =s y J this gives — nat. log y=st + c. When ( = 0, 

* . , (te 

then y = a and c = — nat. log a. Again, let — = a be the 

dt 
Telocity of the point Fy then x = (jU. Substituting for t and c 

their values and remembering that a = 10' and that by defini- 
tion X =s Nap. log y, we get 

10' 
Nap. log y = 10' nat. log — 

if 

It is evident from this formula that Napier's logarithms are 
not the same as the natural logarithms. Napier's logarithms 
increase as the number itself decreases. He took the loga- 
rithm of sin 90 = 0; t.e. the logarithm of 10' = 0. The loga- 
rithm of sin a increased from zero as a decreased from 90^ 
Napier's genesis of logarithms from the conception of two 
flowing points reminds us of Newton's doctrine of fluxions. 
The relation between geometric and arithmetical progressions^ 
so skilfully utilised by Napier, had been observed by Archi- 
medes, Stifel, and others. Napier did not determine the base 
to his system of logarithms. The notion of a '^ base " in fact 
never suggested itself to him. The one demanded by his 
reasoning is the reciprocal of that of the natural system, but 
such a base would not reproduce accurately all of Napier's 
figures, owing to slight inaccuracies in the calculation of the 
tables. Napier's great invention was given to the world in 
1614 in a work entitled Mirifici logarithmorum canonis de- 
9criptio. In it he explained the nature of his logarithms, and 
gave a logarithmic table of the natural sines of a quadrant 
from minute to minute. 

Henry Briggs (1556-1631), in Napier's time professor of 
geometry at Gresham College, London, and afterwards 
professor at Oxford, was so struck with admiration of 
Napier's book, that he left his studies in London to do 



164 A HISTORY OF MATHBliATIGS. 

homage to the Scottish philosopher. Briggs was delayed in 
his journey, and l^apier complained to a common friend, ''Ah, 
John, Mr. Briggs will not come." At that very moment 
knocks were heard at the gate, and Briggs was brought into 
the lord's chamber. Almost one-quarter of an hour was 
spent, each beholding the other without speaking a word. At 
last Briggs began: ''My lord, I have undertaken this long 
journey purposely to see your person, and to know by what 
engine of wit or ingenuity you came first to think of this most 
excellent help in astronomy, viz. the logarithms; but, my 
lord, being by you found out, I wonder nobody found it out 
before, when now known it is so easy." ^ Briggs suggested 
to Napier the advantage that would result from retaining zero 
for the logarithm of the whole sine, but choosing 10,000,000,000 
for the logarithm of the 10th part of that same sine, Le, of 
5® 44' 22". Napier said that he had already thought of the 
change, and he pointed out a slight improvement on Briggs' 
idea; viz. that zero should be the logarithm of 1, and 
10,000,000,000 that of the whole sine, thereby making the 
characteristic of numbers greater than unity positive and 
not negative, as suggested by Briggs. Brjggs admitted this 
to be more convenient. The invention of "Briggian loga- 
rithms" occurred, therefore, to Briggs and Napier indepen- 
dently. The great practical advantage of the new system 
was that its fundamental progression was accommodated to 
the base, 10, of our numerical scale. Briggs devoted all 
his energies to the construction of tables upon the new plan. 
Napier died in 1617, with the satisfaction of having found 
in Briggs an able friend to bring to completion his un- 
finished plans. In 1624 Briggs published his Arithmetica loga- 
rithmica, containing the logarithms to 14 places of numbers, 
from 1 to 20,000 and from 90,000 to 100,000. The gap from 
20,000 to 90,000 was filled up by that illustrious successor of 



^ 



VIETA TO DESCARTES. 166 

Napier and Briggs, Adrian Vlacq of Qouda in Holland. He 
published in 1628 a table of logarithms from 1 to 100,000, of 
which 70,000 were calculated by himself. The first publication 
of Briggian logarithms of trigonometric functions was made 
in 1620 by Gunter, a colleague of Briggs, who found the loga- 
rithmic sines and tangents for every minute to seven places. 
Gunter was the inventor of the words cosine and cotangent. 
Briggs devoted the last years of his life to calculating more 
extensive Briggian logarithms of trigonometric functions, but 
he died in 1631, leaving his work unfinished. It was carried 
on by the English Henry Gellibrand, and then published by 
yiacq at his own expense. Briggs divided a degree into 100 
parts, but owing to the publication by Vlacq of trigonometrical 
tables constructed on the old sexagesimal division, Briggs' 
innovation remained unrecognised. Briggs and Vlacq published 
four fundamental works, the results of which '< have never been 
superseded by any subsequent calculations." 

The first logarithms upon the natural base e were published 
by John Speidell in his New Logarithmea (London, 1619), which 
contains the natural logarithms of sines, tangents, and secants. 

The only possible rival of John Napier in the invention of 
logarithms was the Swiss Justus Byrgius (Joost Biirgi). He 
published a rude table of logarithms six years after the 
appearance of the Canon Mirificua^ but it appears that he 
conceived the idea and constructed that table as early, if not 
earlier, than Napier did his. But he neglected to have the 
results published until Napier's logarithms were known and 
admired throughout Europe. 

Among the various inventions of Napier to assist the 
memory of the student or calculator, is "Napier's rule of 
circular parts" for the solution of spherical right triangles. 
It is, perhaps, "the happiest example of artificial memory 
that is known." 



166 A HISTORY OF MATHEBiATIGS 

The most brilliant conquest in algebra during the sixteenth 
century had been the solution of cubic and bi-quadratic equa- 
tions. All attempts at solving algebraically equations of higher 
degrees remaining fruitless^ a new line of inquiry — the prop- 
erties of equations and their roots — was gradually opened up. 
We have seen that Vieta had attained a partial knowledge of 
the relations between roots and coefficients. Peletarios, a 
Frenchman, had observed as early as 1558, that the root of an 
equation is a divisor of the last term. One who extended the 
theory of equations somewhat further than Vieta, was Albert 
Girard (1590-1634), a Flemish mathematician. Like Vieta, 
this ingenious author applied algebra to geometry, and was 
the first who understood the use of negative roots in the 
solution of geometric problems. He spoke of imaginary quan- 
tities ; inferred by induction that every equation has as many 
roots as there are units in the number expressing its degree ; 
and first showed how to express the sums of their powers in 
terms of the coefficients. Another algebraist of considerable 
power was the English Thomas Harriot (1560-1621). He 
accompanied the first colony sent out by Sir Walter Kaleigh 
to Virginia. After having surveyed that country he returned 
to England. As a mathematician, he was the boast of his 
country. He brought the theory of equations under one 
comprehensive point of view by grasping that truth in its 
full extent to which Vieta and Girard only approximated ; viz. 
that in an equation in its simplest form, the coefficient of 
the second term with its sign changed is equal to the sum of 
the roots ; the coefficient of the third is equal to the sum 
of the products of every two of the roots ; etc. He was the 
first to decompose equations into their simple factors; but, 
since he failed to recognise imaginary and even negative roots, 
he failed also to prove that every equation could be thus 
decomposed. Harriot made some changes in algebraic nota- 



VIBTA TO DESCARTES. 167 

tion, adopting small letters of the alphabet in place of the 
capitals used by Vieta. The symbols of inequality > and < 
were introduced by him. Harriot's work, Artis AnalyticfB 
praxis, was published in 1631, ten years after his death. 
William Oaghtred (1574-1660) contributed vastly to the propa- 
gation of mathematical knowledge in England by his treatises, 
which were long used in the universities. He introduced x 
as symbol of multiplication, and : : as that of proportion. By 
him ratio was expressed by only one dot. In the eighteenth 
century Christian Wolf secured the general adoption of the 
dot as a symbol of multiplication, and the sign for ratio was 
thereupon changed to two dots. Oughtred's ministerial duties 
left him but little time for the pursuit of mathematics during 
daytime, and evenings his economical wife denied him the 
use of a light. 

Algebra was now in a state of sufficient perfection to enable 
Descartes to take that important step which forms one of the 
grand epochs in the history of mathematics, — the application 
of algebraic analysis to define the nature and investigate the 
properties of algebraic curves. 

In geometry, the determination of the areas of curvilinear 
figures was diligently studied at this period. Paul Goldin 
(1577-1643), a Swiss mathematician of considerable note, 
rediscovered the following theorem, published in his Centra- 
haryca, which has been named after him, though first found 
in the Mathematical Collections of Pappus: The volume of a 
solid of revolution is equal to the area of the generating 
figure, multiplied by the circumference described by the centre 
of gravity. We shall see that this method excels that of 
Kepler and Cavalieri in following a more exact and natural 
course ; but it has the disadvantage of necessitating the deter- xV 
mination of the centre of gravity, which in itself may be a 
more difficult problem than the original one of finding the 



168 A HISTORY OF MATHEMATICS. 

Yolmne. Guldin made some attempts to prove his theorem^ 
but Gavalieri pointed out the weakness of his demonstration. 

Johannes Kepler (1671-1630) was a native of Wdrtemberg 
and imbibed Copernican principles while at the University of 
Tubingen. His pursuit of science was repeatedly interrupted 
by war, religious persecution, pecuniary embarrassments, fre- 
quent changes of residence, and family troubles. In 1600 he 
became for one year assistant to the Danish astronomer, Tycho 
Brahe, in the observatory near Prague. The relation between 
the two great astronomers was not always of an ^^eeable 
character. Kepler's publications are voluminous. His first 
attempt to explain the solar system was made in 1596, when he 
thought he had discovered a ciirious relation between the 
five regular solids and the number and distance of the planets. 
The publication of this pseudo-discovery brought him much 
fame. Maturer reflection and intercourse with Tycho Brahe 
and Galileo led him to investigations and results more worthy of 
his genius — "Kepler's laws." He enriched pure mathematics 
as well as astronomy. It is not strange that he was interested 
in the mathematical science which had done him so much 
service ; for " if the Greeks had not cultivated conic sections, 
Kepler could not have superseded Ptolemy."" The Greeks 
never dreamed that these curves would ever be of practical 
use ; Aristaeus and ApoUonius studied them merely to satisfy 
their intellectual cravings after the ideal ; yet the conic 
sections assisted Kepler in tracing the march of the planets in 
their elliptic orbits. Kepler made also extended use of logar 
rithms and decimal fractions, and was enthusiastic in diffusing 
a knowledge of them. At one time, while purchasing wine, he 
was struck by the inaccuracy of the ordinary modes of deter- 
mining the contents of kegs. This led him to the study of 
the volumes of solids of revolution and to the publication of 
the Stereometria Doliorum in 1615. In it he deals first with the 



VIETA TO DESCABTES. 169 

solids known to Archimedes and then takes up others. Kepler 
introduced a new idea into geometry ; namely, that of infinitely 
great and infinitely small quantities. Greek mathematicians 
always shunned this notion, but with it modern mathematicians 
have completely revolutionised the science. In comparing 
rectilinear figures, the method of superposition was employed 
by the ancients, but in comparing rectilinear and curvilinear 
figures with each other, this method failed because no addition 
or subtraction of rectilinear figures could ever produce curvi- 
linear ones. To meet this case, they devised the Method of 
Exhaustion, which was long and difficult; it was purely 
synthetical, and in general required that the conclusion should 
be known at the outset. The new notion of infinity ledl 
gradually to the invention of methods immeasurably more* 
powerful Kepler conceived the circle to be composed of an 
infinite number of triangles having their common vertices at 
the centre, and their bases in the circumference; and the 
sphere to consist of an infinite number of pyramids. He 
applied conceptions of this kind to the determination of the 
areas and volumes of figures generated by curves revolving 
about any line as axis, but succeeded in solving only a few of 
the simplest out of the 84 problems which he proposed for 
investigation in his Stereometria. 

Other points of mathematical interest in Kepler's works are 
(1) the statement of the earliest problem of inverse tan- 
gents ; (2) an investigation which amounts to the evaluation of 

the definite integral I sin <f>d<f> = 1 — cos <^ ; (3) the assertion 

that the circumference of an ellipse, whose axes are 2 a and 
2b f is nearly ir (a + 6) ; (4) a passage from which it has been 
inferred that Kepler knew the variation of a function near its 
maximum value to disappear ; (5) the assumption of the prin- 
ciple of continuity (which differentiates modern from ancient 
geometry), when he shows that a parabola has a focus at 



170 A HISTOEV OF MATHEMATICS. 

infinity, that lines radiating from this ^'csBCus focus" are 
parallel and have no other point at infinity. 

The Stereometria led Gavalieri, an Italian Jesuit, to the 
consideration of infinitely small quantities. Bonaveiitura 
Cavalieri (1598-1647), a pupil of Galileo and professor at 
Bologna, is celebrated for his Oeometria indivisibilibus con- 
tinuorum nova quadam ratione promota, 1635. This work 
expounds his method of Indivisibles, which occupies an inter- 
mediate place between the method of exhaustion of the Greeks 
and the methods of Newton and Leibniz. He considers lines 
as composed of an infinite number of points, surfaces as com- 
posed of an infinite number of lines, and solids of an infinite 
number of planes. The relative magnitude of two solids or 
surfaces could then be found simply by the summation of 
series of planes or lines. For example, he finds the sum of the 
squares of all lines making up a triangle equal to one-third 
the sum of the squares of all lines of a parallelogram of equal 
base and altitude ; for if in a triangle, the first line at the apex 
be 1, then the second is 2, the third is 3, and so on ; and the 
sum of their squares is 

12 _|. 22 -h 32 -h ••• -h n^ = n (n + 1) (2n -h 1) -i- 6. 

In the parallelogram, each of the lines is n and their number is 
n; hence the total sum of their squares is n\ The ratio 
between the tnfo sums is therefore 

n(n + l)(2n-hl)^67i» = i, 

since n is infinite. From this he concludes that the pyramid 
or cone is respectively ^ of a prism or cylinder of equal base 
and altitude, since the polygons or circles composing the for- 
mer decrease from the base to the apex in the same way as the 
squares of the lines parallel to the base in a triangle decrease 
from base to apex. By the Method of Indivisibles, Cavalieri 



VIBTA TO DESCARTES. 171 

solved the majority of the problems proposed by Kepler. 
Though expeditious and yielding correct results, Cavalieri's 
method lacks a scientific foundation. If a line has absolutely 
no width, then no number, however great, of lines can ever 
make up an area ; if a plane has no thickness whatever, then 
even an infinite number of planes cannot form a solid. The 
reason why this method led to correct conclusions is that one 
area is to another area in the same ratio as the sum of the 
series of lines in the one is to the sum of the series of lines in 
the other. Though unscientific, Cavalieri's method was used 
for fifty years as a sort of integral calculus. It yielded solu- 
tions to some difficult problems. Guldin made a severe attack 
on Cavalieri and his method. The latter published in 1647, 
after the death of Guldin, a treatise entitled ExercUationes 
geametricoB sex, in which he replied to the objections of his 
opponent and attempted to give a clearer explanation of his 
method. Guldin had never been able to demonstrate the 
theorem named after him, except by metaphysical reasoning, 
but Cavalieri proved it by the method of indivisibles. A 
revised edition of the Geometry of Indivisibles appeared in 
1653. 

There is an important curve, not known to the ancients, 
which now began to be studied with great zeal. Roberval 
gave it the name of " trochoid," Pascal the name of " roulette," 
Galileo the name of " cycloid." The invention of this curve 
seems to be due to Galileo, who valued it for the graceful form 
it would give to arches in architecture. He ascertained its 
area by weighing paper figures of the cycloid against that of 
the generating circle, and found thereby the first area to be 
nearly but not exactly thrice the latter. A mathematical 
determination was made by his pupil, Evangelista Torricelli 
(1608-1647), who is more widely known as a physicist than 
as a mathematician. 



172 A HISTORY OF MATHEMATICS. 

By the Method of Indivisibles he demonstrated its area to 
be triple that of the revolving circle, and published his 
solution. This same quadrature had been effected a few years 
earlier by Koberval in France, but his solution was not known 
to the Italians. Koberval, being a man of irritable and 
violent disposition, unjustly accused the mild and amiable 
Torricelli of stealing the proof. This accusation of plagiarism 
created so much chagrin with Torricelli that it is considered 
to have been the cause of his early death. Vincenzo Viviaiiii 
another prominent pupil of Galileo, determined the tangent 
to the cycloid. This was accomplished in France by Descartes 
and Fermat. 

In France, where geometry began to be cultivated with 

' greatest success, Roberval, Fermat, Pascal, employed the 
Method of Indivisibles and made new improvements in it. 
Giles Persone de Roberval (1602-1675), for forty years pro- 
fessor of mathematics at the College of France in Paris, 
claimed for himself the invention of the Method of Indivis- 
ibles. Since his complete works were not published until 
after his death, it is difficult to settle questions of priority. 
Montucla and Chasles are of the opinion that he invented the 
method independent of and earlier than the Italian geometer, 
though the work of the latter was published much earlier 
than Roberval's. Marie finds it difficult to believe that the 
Frenchman borrowed nothing whatever from the Italian, for 
both could not have hit independently upon the word Indi- 
visibles, which is applicable to infinitely small quantities, as 

r conceived by Cavalieri, but not as conceived by Roberval. 
Koberval and Pascal improved the rational basis of the 
Method of Indivisibles, by considering an area as 'made up 
of an indefinite number of rectangles instead of lines, and 
a solid as composed of indefinitely small solids instead of 
surfaces. Roberval applied the method to the finding of 



VIBTA TO DESGABTBS. 178 

areaSy volameS; and centres of gravity. He effected the quad- 
rature of a parabola of any degree y* = a'^'^x, and also of a 
parabola ^ = a^'^af*. We have already mentioned his quadra- 
ture of the cycloid. Koberval is best known for his method 
of drawing tangents. He was the first to apply motion to 
the resolution of this important problem. His method is 
allied to Kewton's principle of fluxions. Archimedes con- 
ceived his spiral to be generated by a double motion. This 
idea Roberval extended to all curves. Plane curves, as for 
instance the conic sections, may be generated by a point 
acted upon by two forces, and are the resultant of two 
motions. If at any point of the curve the resultant be 
resolved into its components, then the diagonal of the par- 
allelogram determined by them is the tangent to the curve 
at that point. The greatest difficulty connected with this 
ingenious method consisted in resolving the resultant into 
components having the proper lengths and directions. Kober- 
val did not always succeed in doing this, yet his new idea was 
a great step in advance. He broke ofE from the ancient 
definition of a tangent as a straight line having only one point 
in common with a curve, — a definition not valid for curves of 
higher degrees, nor apt even in curves of the second degree to 
bring out the properties of tangents and the parts they may 
be made to play in the generation of the curves. The subject 
of tangents received special attention also from Fermat, 
Descartes, and Barrow, and reached its highest development 
after the invention of the differential calculus. Fermat and 
Descartes defined tangents as secants whose two points of 
iiktefseetiOiPwIth" the curve coincide; Barrow considered a 
curre a polygon, and called one of its sides produced a tangent. 
A profound scholar in all branches of learning and a mathe- 
matician of exceptional powers was Pierre de Fermat (1601- 
1665). He studied law at Toulouse, and in 1631 was made 



\ 



174 A HISTORY OF MATHEMATICS. 

councillor for the parliament of Toulouse. His leisure time 
was mostly devoted to mathematics, which he studied with 
irresistible passion. Unlike Descartes and Pascal, he led a 
quiet and unaggressive life. Fermat has left the impress of 
his genius upon all branches of mathematics then known. A 
great contribution to geometry was his De maximis et minimis. 
About twenty years earlier, Kepler had first observed that the 
increment of a variable, as, for instance, the ordinate of a 
curve, is evanescent for values very near a maximum or a 
minimum value of the variable. Developing this idea, Fermat 
obtained his rule for maxima and minima. He substituted 
x-\-e for X in the given function of x and then equated to each 
other the two consecutive values of the function and divided 
the equation by e. If e be taken 0, then the roots of this 
equation are the values of x, making the function a maximum 
or a minimum. Fermat was in possession of this rule in 1629. 
The main difference between it and the rule of the differential 
calculus is that it introduces the indefinite quantity e instead 
of the infinitely small dx. Fermat made it the basis for his 
method of drawing tangents. 

Owing to a want of explicitness in statement, Fermat's 
method of maxima and minima, and of tangents, was severely 
attacked by his great contemporary, Descartes, who could 
never be brought to render due justice to his merit. In the 
ensuing dispute, Fermat found two zealous defenders in Rober- 
val and Pascal, the father; while Midorge, Desargues, and 
Hardy supported Descartes. 

Since Fermat introduced the conception of infinitely small 
differences between consecutive values of a function and ar- 
rived at the principle for finding the maxima and minima, 
it was maintained by Lagrange, Laplace, and Fourier, that 
Fermat may be regarded as the first inventor of the differ- 
ential calculus. This point is not well taken, as will be seen 



VIETA TO DESCABTE8. 176 

from the words of Poisson, himself a Frenchman, who rightly 
says that the differential calculus '' consists in a system of 
rules proper for finding the differentials of all functions, 
rather than in the use which may be made of these infinitely 
small variations in the solution of one or two isolated prob- 
lems." 

A contemporary mathematician, whose genius excelled even 
that of the great Fermat, was BUise Pascal (1623-1662). He 
was born at Clermont in Auvergne. In 1626 his father retired 
to Paris, where he devoted himself to teaching his son, for he 
would not trust his education to others. Blaise Pascal's genius 
for geometry showed itself when he was but twelve years 
old. His father was well skilled in mathematics, but did not 
wish his son to study it until he was perfectly acquainted with 
Latin and Greek. All mathematical books were hidden out of 
his sight. The boy once asked his father what mathematics 
treated of, and was answered, in general, ''that it was the 
method of making figures with exactness, and of finding out 
what proportions they relatively had to one another." He 
was at the same time forbidden to talk any more about it, 
or ever to think of it. But his genius could not submit to be 
confined within these bounds. Starting with the bare fact 
that mathematics taught the means of making figures infalli- 
bly exact, he employed his thoughts about it and with a piece 
of charcoal drew figures upon the tiles of the pavement, trying 
the methods of drawing, for example, an exact circle or equi- 
lateral triangle. He gave names of his own to these figures 
and then formed axioms, and, in short, came to make perfect 
demonstrations. In this way he arrived unaided at the theo- 
rem that the sum of the three angles of a triangle is equal to 
two right angles. His father caught him in the act of study- 
ing this theorem, and was so astonished at the sublimity and 
force of his genius as to weep for joy. The father now gave 



176 A HISTORY OF MATHEMATICS. 

him Euclid's Elements, which he, without assistance, mastered 
easily. His regular studies being languages, the boy employed 
only his hours of amusement on the study of geometry, yet he 
had so ready and lively a penetration that, at the age of six- 
teen, he wrote a treatise upon conies, which passed for such a 
surprising effort of genius, that it was said nothing -equal to it 
in strength had been produced since the time of Archimedes. 
Descartes refused to believe that it was written by one so 
young as Pascal. This treatise was never published, and is 
now lost. Leibniz saw it in Paris and reported on a portion 
of its contents. The precocious youth made vast progress in 
all the sciences, but the constant application at so tender an 
age greatly impaired his health. Yet he continued working, 
and at nineteen invented his famous machine for performing 
arithmetical operations mechanically. This continued strain 
from overwork resulted in a permanent indisposition, and he 
would sometimes say that from the time he was eighteen, 
he never passed a day free from pain. At the age of 
twenty-four he resolved to lay aside the study of the human 
sciences and to consecrate his talents to religion. His Pro- 
vincial Letters against the Jesuits are celebrated. But at 
times he returned to the favourite study of his youth. Being 
kept awake one night by a toothache, some thoughts un- 
designedly came into his head concerning the roulette or 
cycloid; one idea followed another; and he thus discovered 
properties of this curve even to demonstration. A corre- 
spondence between him and Fermat on certain problems was 
the beginning of the theory of probability. Pascal's illness 
increased, and he died at Paris at the early age of thirty-nine 
years.*^ By him the answer to the objection to Cavalieri's 
Method of Indivisibles was put in the clearest form. Like 
Koberval, he explained " the sum of right lines " to mean " the 
sum of infinitely small rectangles." Pascal greatly advanced 



VIBTA TO DESCARTES. 177 

the knowledge of the cycloid. He determined the area of a 
section produced by any line parallel to the base ; the volume 
generated by it revolving around its base or around the axis ; 
and, finally, the centres of gravity of these volumes, and also 
of half these volumes cut by planes of symmetry. Before 
publishing his results, he sent, in 1658, to all mathematicians 
that famous challenge offering prizes for the first two solu- 
tions of these problems. Only Wallis and A. La Louvre com- 
peted for them. The latter was quite unequal to the task; 
the former, being pressed for time, made numerous mistakes : 
neither got a prize. Pascal then published his own solutions, 
which produced a great sensation among scientific men. Wal- 
lis, too, published his, with the errors corrected. Though not 
competing for the prizes, Huygens, Wren, and Format solved 
some of the questions. The chief discoveries of Christopher 
Wren (1632-1723), the celebrated architect of St. Paul's 
Cathedral in London, were the rectification of a cycloidal arc 
and the determination of its centre of gravity. Fermat 
found the area generated by an arc of the cycloid. Huygens 
invented the cycloidal pendulum. 

The beginning of the seventeenth century witnessed also 
a revival of synthetic geometry. One who treated conies still 
by ancient methods, but who succeeded in greatly simplifying 
many prolix proofs of Apollonius, was Claude Mydorge in 
Paris (1585-1647), a friend of Descartes. But it remained 
for Girard Desargues (1593-1662) of Lyons, and for Pascal, to 
leave the beaten track and cut out fresh paths. They intro- 
duced the important method of Perspective. All conies on 
a cone with circular base appear circular to an eye at the apex. 
Hence Desargues and Pascal conceived the treatment of the 
conic sections as projections of circles. Two important and 
beautiful theorems were given by Desargues : The one is on 
the '^involution of the six points," in which a transversal 



178 A HISTORY OF MATHEMATICS. 

meets a conic and an inscribed quadrangle ; the other is that, 
if the vertices of two triangles, situated either in space or in 
a plane, lie on three lines meeting in a point, then their sides 
meet in three points lying on a line; and conversely. This 
last theorem has been employed in recent times by Branchion, 
Sturm, Gergonne, and Poncelet. Poncelet made it the basis 
of his beautiful theory of homoligical figures. We owe to 
Desargues the theory of involution and of transversals ; also 
the beautiful conception that the two extremities of a straight 
line may be considered as meeting at infinity, and that paral- 
lels differ from other pairs of lines only in having their points 
of intersection at infinity. Pascal greatly admired Desargues' 
results, saying (in his Essais pour les Coniques), " I wish to 
acknowledge that I owe the little that I have discovered on 
this subject, to his writings." Pascal's and Desargues' writ- 
ings contained the fundamental ideas of modern synthetic 
geometry. In Pascal's wonderful work on conies, written 
at the age of sixteen and now lost, were given the theorem 
on the anharmonic ratio, first found in Pappus, and also that 
celebrated proposition on the mystic hexagon, known as 
" Pascal's theorem," viz. that the opposite sides of a hexagon 
inscribed in a conic intersect in three points which are col- 
linear. This theorem formed the keystone to his theory. He 
himself said that from this alone he deduced over 400 corol- 
laries, embracing the conies of Apollonius and many other 
results. Thus the genius of Desargues and Pascal uncovered 
several of the rich treasures of modem synthetic geometry; 
but owing to the absorbing interest taken in the analytical 
geometry of Descartes and later in the differential calculus, 
the subject was almost entirely neglected until the present 
century. 

In the theory of numbers no new results of scientific value 
had been reached for over 1000 years, extending from the 



VIETA TO DESCARTES. 179 

times of Diophantus and the Hindoos until the beginning of 
the seventeenth century. But the illustrious period we are 
now considering produced men who rescued this science from 
the realm of mysticism and superstition, in which it had been 
so long imprisoned; the properties of numbers began again 
to be studied scientifically. Not being in possession of the 
Hindoo indeterminate analysis, many beautiful results of the 
Brahmins had to be re-discovered by the Europeans. Thus 
a solution in integers of linear indeterminate equations was 
le-discovered by the Frenchman Bachet de Meziriac (1581- 
1638), who was the earliest noteworthy European Diophantist. 
In 1612 he published ProbUmes plaiaants et d4lectables qui se 
font par lea nombresy and in 1621 a Greek edition of Diophantus 
with notes. The father of the modem theory of numbers is 
Fermat. He was so uncommunicative in disposition, that he 
generally concealed his methods and made known his resiQts 
only. In some cases later analysts have been greatly puzzled 
in the attempt of supplying the proofs. Fermat owned a copy 
of Bachet's Diophantus, in which he entered numerous mar- 
ginal notes. In 1670 these notes were incorporated in a new 
edition of Diophantus^ brought out by his son. Other theorems 
on numbers, due to Fermat, were published in his Opera vana 
(edited by his son) and in Wallis's Commercium epistolicum 
of 1658. Of the following theorems, the first seven are found 
in the marginal notes : — 

(1) a5*-|-y" = z" is impossible for integral values of a?, y, 
and 2, when n > 2. Remark : " I have found for this a truly 
wonderful proof, but the margin is too small to hold it." 
Repeatedly was this theorem made the prize question of 
learned societies. It has given rise to investigations of 
great interest and difficulty on the part of Euler, Lagrange, 
Dirichlet, and Kummer. 

(2) A prime of the form 4 n+l is only once the hypothenuse 



180 A HISTORY OF MATHEMATICS. 

of a right triangle; its square is twice; its cube is three 
ti ines, etc. Example : 6« = 3* -h 4* ; 25« = 15* -h 20* = 7« + 24^ ; 
125« = 75* + 100^ = 35* + 120* = 44* -h 117*. 

(3) A prime of the form 4 n + 1 can be expressed once, and 
only once, as the sum of two squares. Proved by Euler. 

(4) A number composed of two cubes can be resolved into 
two other cubes in an infinite multiplicity of ways. 

(5) Every number is either a triangular number or the sum 
of two or three triangular numbers; either a square or the 
sum of two, three, or four squares ; either a pentagonal number 
or the sum of two, three, four, or five pentagonal numbers; 
similarly for polygonal numbers in general. The proof of this 
and other theorems is promised by Fermat in a future work 
which never appeared. This theorem is also given, with 
others, in a letter of 1637 (?) addressed to Pater Mersenne. 

(6) As many numbers as you please may be found, such 
that the square of each remains a square on the addition to 
or subtraction from it of the sum of all the numbers. 

(7) X* -{- y*= z^ is impossible. 

(8) In a letter of 1640 he gives the celebrated theorem 
generally known as "Fermat's theorem," which we state in 
Gauss's notation : If p is prime, and a is prime to p, then af~' 
= 1 (modp). It was proved by Euler. 

(9) Fermat died with the belief that he had found a long- 
sought-for law of prime numbers in the formula 2* -|- 1 = a 
prime, but he admitted that he was unable to prove it rigor- 
ously. The law is not true, as was pointed out by Euler in the 
example 2^*4- 1 = 4,294,967,297 = 6,700,417 times 641. The 
American liglitniug calculator Zerah Colbuim, when a boy, 
readily found the factors, but was unable to explain the 
method by which he niaile his marvellous mental computation. 

(10) An odd prime number can be expressed as the differ- 
ence of two squares in one, and only one, way. This theorem. 



VIBTA TO DESCARTES. 181 

given in the Belation, was used by Fermat for the decomposi- 
tion of large numbers into prime factors. 

(11) If the integers a, 6, c represent the sides of a right 
triangle^ then its area cannot be a square number. This was 
proved by Lagrange. 

(12) Fermat's solution of aa:^ -|- 1 = y*, where a is integral 
but not a square, has come down in only the broadest outline, 
as given in the Relation, He proposed the problem to the 
Frenchman, Bemhard Frenide de Bessy, and in 1657 to all 
living mathematicians. In England, Wallis and Lord Brounker 
conjointly found a laborious solution, which was published in 
1658, and also in 1668, in an algebraical work brought out by 
John Pell, Though Pell had no other connection with the 
problem, it went by the name of " PelPs problem." The first 
solution was given by the Hindoos. 

We are not sure that Fermat subjected all his theorems to 
rigorous proof. His methods of proof were entirely lost 
until 1879, when a document was found buried among the 
manuscripts of Huygens in the library of Leyden, entitled 
Relation des dicouvertes en la science des nombres. It appears 
from it that he used an inductive method, called by him la 
descente infinie ou indejinie. He says that this was particularly 
applicable in proving the impossibility of certain relations, as, 
for instance. Theorem 11, given above, but that he succeeded 
in using the method also in proving affirmative statements. 
Thus he proved Theorem 3 by showing that if we suppose 
there be a prime 4 n -h 1 which does riot possess this property, 
then there will be a smaller prime of the form 4 n -h 1 not 
possessing it; and a third one smaller than the second, not 
possessing it; and so on. Thus descending indefinitely, he 
arrives at the number 5, which is the smallest prime factor 
of the form 4 n -|- 1. From the above supposition it would 
follow that 5 is not the sum of two squares — a conclusion 



182 A HISTORY OF MATHBliATICS. 

contrary to fact. Hence the supposition is false, and the 
theorem is established. Fermat applied this method of de- 
scent with success in a large number of theorems. By this 
method Euler, Legendre, Dirichlet, proved several of his 
enunciations and many other numerical propositions. 

A correspondence between Pascal and Fermat relating to a 
certain game of chance was the germ of the theory of proba- 
bilitieS; which has since attained a vast growth. Chevalier de 
M^r^ proposed to Pascal the fundamental problem, to deter- 
mine the probability which each player has, at any given 
stage of the game, of winning the game. Pascal and Fermat 
supposed that the players have equal chances of winning a 
single point. 

The former communicated this problem to Fermat, who 
studied it with lively interest and solved it by the theory of 
combinations, a theory which was diligently studied both by 
him and Pascal. The calculus of probabilities engaged the 
attention also of Huygens. The most important theorem 
reached by him was that, if A has p chances of winning a sum 
a, and q chances of winning a sum b, then he may expect to 
win the sum ?5P_±L?2. The next great work on the theory of 

probability was the Ars conjectandi of Jakob Bernoulli. 

Among the ancients, Archimedes was the only one who 
attained clear and correct notions on theoretical statics. He 
had acquired firm possession of the idea of pressure, which 
lies at the root of mechanical science. But his ideas slept 
nearly twenty centuries, until the time of Stevin and Galileo. 
Stevin determined accurately the force necessary to sustain 
a body on a plane inclined at any angle to the horizon. He 
was in possession of a complete doctrine of equilibrium. While 
Stevin investigated statics, Galileo pursued principally dynam- 
ics. Galileo was the first to abandon the Aristotelian idea 
that bodies descend more quickly in proportion as they are 



DESCARTES TO NEWTON. 188 

heavier; he established the first law of motion; determined 
the laws of falling bodies ; and, having obtained a clear notion 
of acceleration and of the independence of different motions, 
was able to prove that projectiles move in parabolic curves. 
Up to his time it was believed that a cannon-ball moved 
forward at first in a straight line and then suddenly fell 
vertically to the ground. Galileo had an understanding of 
centrifugal forces, and gave a correct definition of momentum. 
Though he formulated the fundamental principle of statics, 
known as the parallelogram of forces, yet he did not fully 
recognise its scope. The principle of virtual velocities was 
partly conceived by Guido Ubaldo (died 1607), and afterwards 
more fully by Galileo. 

Galileo is the founder of the science of dynamics. Among 
his contemporaries it was chiefly the novelties he detected in 
the sky that made him celebrated, but Lagrange claims that 
his astronomical discoveries required only a telescope and 
perseverance, while it took an extraordinary genius to dis- 
cover laws from phenomena, which we see constantly and of 
which the true explanation escaped all earlier philosophers. 
The first contributor to the science of mechanics after Gralileo 
was Descartes. 



DESCARTES TO NEWTON. 

Among the earliest thinkers of the seventeenth and eigh- 
teenth centuries, who employed their mental powers toward the 
destruction of old ideas and the up-building of new ones, ranks 
Rene Descartes (1596-1650). Though he professed orthodoxy 
in faith all his life, yet in science he was a profound sceptic. 
He found that the world's brightest thinkers had been long 
exercised in metaphysics, yet they had discovered nothing 



184 A HISTORY OP MATHEMATICS. 

certain; nay, had even flatly contradicted each other. This 
led him to the gigantic resolution of taking nothing whatever 
on authority, but of subjecting everything to scrutinous exam- 
ination, according to new methods of inquiry. The certainty 
of the conclusions in geometry and arithmetic brought out in 
his mind the contrast between the true and false ways of 
seeking the truth. He thereupon attempted to apply mathe- 
matical reasoning to all sciences. ** Comparing the mysteries 
of nature with the laws of mathematics, he dared to hope that 
the secrets of both could be unlocked with the same key." 
Thus he built up a system of philosophy called Cartesianism. 
Great as was Descartes' celebrity as a metaphysician, it 
may be fairly questioned whether his claim to be remembered 
by posterity as a mathematician is not greater. His philosophy 
has long since been superseded by other systems, but the ana- 
lytical geometry of Descartes will remain a valuable possession 
forever. At the age of twenty-one, Descartes enlisted in the 
army of Prince Maurice of Orange. His years of soldiering 
were years of leisure, in which he had time to pursue his 
studies. At that time mathematics was his favourite science. 
But in 1625 he ceased to devote himself to pure mathematics. 
Sir William Hamilton is in error when he states that Descartes 
considered mathematical studies absolutely pernicious as a 
means of internal culture. In a letter to Mersenne, Descartes 
says : " M. Desargues puts me under obligations on account of 
the pains that it has pleased him to have in me, in that he 
shows that he is sorry that I do not wish to study more in geom- 
etry, but I have resolved to quit only abstract geometry, that 
is to say, the consideration of questions which serve only to 
exercise the mind, and this, in order to study another kind of 
geometry, which has for its object the explanation of the 
phenomena of nature. . . . You know that all my physics is 
nothing else than geometry." The years between 1629 and 



DESCARTES TO NEWTON. 185 

1649 were passed by him in Holland in the study, principally, 
of physics and metaphysics. His residence in Holland was 
during the most brilliant days of the Dutch state. In 1G37 he 
published his Discours de la Mithod£, containing among others 
an essay of 106 pages on geometry. His Geometry is not easy 
reading. An edition appeared subsequently with notes by his 
friend De Beaune, which were intended to remove the dif- 
ficulties. 

It is frequently stated that Descartes was the first to apply 
algebra to geometry. This statement is inaccurate, for Vieta 
and others had done this before him. Even the Arabs some- 
times used algebra in connection with geometry. The new 
step that Descartes did take was the introduction into geom- 
etry of an analytical method based on the notion of variables 
and constants, which enabled him to represent curves by alge- 
braic equations. In the Greek geometry, the idea of motion 
was wanting, but with Descartes it became a very fruitful 
conception. By him a point on a plane was determined in 
position by its distances from two fixed right lines or axes. 
These distances varied with every change of position in 
the point. Tliis geometric idea of coordinate representation^ 
together with the algebraic idea of two variables in one equa- 
tion having an indefinite number of simultaneous values, fur- 
nished a method for the study of loci, which is admirable for 
the generality of its solutions. Thus the entire conic sections 
of Apollonius is wrapped up and contained in a single equa- 
tion of the second degree. 

The Latin term for "ordinate," used by Descartes comes 
from the expression lineoR ordinatce, employed by Roman sur- 
veyors for parallel lines. The term abscissa occurs for the 
first time in a Latin work of 1659, written by Stefano degli 
Angeli (1623-1697), a professor of mathematics in Rome.* 
Descartes' geometry was called "analytical geometry," partly 



186 A HISTORY OF MATHBliATICS. 

because, xmlike the synthetic geometry of the ancients, it is 
actually ancUyticcUy in the sense that the word is used in 
logic ; and partly because the practice had then already 
arisen, of designating by the term analysis the calculus with 
general quantities. 

The first important example solved by Descartes in his 
geometry is the "problem of Pappus"; viz. "Given several 
straight lines in a plane, to find the locus of a point such that 
the perpendiculars, or more generally, straight lines at given 
angles, drawn from the point to the given lines, shall satisfy 
the condition that the product of certain of them shall be in 
a given ratio to the product of the rest." Of this celebrated 
problem, the Greeks solved only the special case when the 
number of given lines is four, in which case the locus of the 
point turns out to be a conic section. By Descartes it was 
solved completely, and it afforded an excellent example of 
the use which can be made of his analytical method in the 
study of loci. Another solution was given later by Newton 
in the Principia. 

Tlie methods of drawing tangents invented by Roberval 
and Fermat were noticed earlier. Descartes gave a third 
method. Of all the problems which he solved by his geometry, 
none gave him as great pleasure as his mode of constructing 
tangents. It is profound but operose, and, on that account, 
inferior to Fermat's. His solution rests on the method of 
Indetemiinaie Coefficients, of which he bears the honour of 
invention. Indeterminate coefficients were employed by him 
also in solving bi-quadratic equations. 

The essays of Descartes on dioptrics and geometry were 
sharply criticised by Fermat, who wrote objections to the 
former, and sent his own treatise on "maxima and minima" 
to show that there were omissions in the geometry. Descartes 
thereupon made an attack on Fermat's method of tangents. 



DESCARTES TO NEWTON. 187 

Descartes was in the wrong in this attack, yet he continued 
the controversy with obstinacy. He had a controversy also 
with Roberval on the cycloid. This curve has been called 
the " Helen of geometers," on account of its beautiful proper- 
ties and the controversies which their discovery occasioned. 
Its quadrature by Roberval was generally considered a brill- 
iant achievement, but Descartes commented on it by saying 
that any one moderately well versed in geometry might have 
done this. He then sent a short demonstration of his own. 
On Roberval's intimating that he had been assisted by a 
knowledge of the solution, Descartes constructed the tangent 
to the curve, and challenged Roberval and Fermat to do the 
same. Fermat accomplished it, but Roberval never succeeded 
in solving this problem, which had cost the genius of Des- 
cartes but a moderate degree of attention. 

He studied some new curves, now called "ovals of Des- 
cartes," which were intended by him to serve in the con- 
struction of converging lenses, but which yielded no results 
of practical value. 

The application of algebra to the doctrine of curved lines 
reacted favourably upon algebra. As an abstract science, 
Descartes improved it by the systematic use of exponents and 
by the full interpretation and construction of negative quanti- 
ties. Descartes also established some theorems on the theory of 
equations. Celebrated is his " rule of signs " for determining 
the number of positive and negative roots ; viz. an equation 
may have as many -|- roots as there are variations of signs, and 
as many — roots as there are permanencies of signs. Descartes 
was charged by Wallis with availing himself, without acknowl- 
edgment, of Harriot's theory of equations, particularly his mode 
of generating equations ; but there seems to be no good ground 
for the charge. Wallis also claimed that Descartes failed to 
observe that the above rule of signs is not true whenever the 



188 A HISTORY OF MATHEMATICS. 

equation has imaginary roots ; but Descartes does not say that 
the equation always ha^y but that it may have so many roots. 
It is true that Descartes does not consider the case of imagi- 
naries directly, but further on in his Geometry he gives 
incontestable evidence of being able to handle this case 
also. 

In mechanics, Descartes can hardly be said to have advanced 
beyond Galileo. The latter had overthrown the ideas of 
Aristotle on this subject, and Descartes simply " threw himself 
upon the enemy" that had already been "put to the rout." 
His statement of the first and second laws of motion was an 
improvement in form, but his third law is false in substance. 
The motions of bodies in their direct impact was imperfectly 
understood by Galileo, erroneously given by Descartes, and 
first correctly stated by Wren, Wallis, and Huygens. 

One of the most devoted pupils of Descartes was the learned 
Pnncess Elizabeth, daughter of Frederick V. She applied the 
new analytical geometry to the solution of the " Apollonian 
problem." His second royal follower was Queen Christina, 
the daughter of Gustavus Adolphus. She urged upon Des- 
cartes to come to the Swedish court. After much hesitation 
he accepted the invitation in 1649. He died at Stockholm one 
year later. His life had been one long warfare against the 
prejudices of men. 

It is most remarkable that the mathematics and philosophy 
of Descartes should at first have been appreciated less by his 
countrymen than by foreigners. The indiscreet temper of 
Descartes alienated the great contemporary French mathema- 
ticians, Roberval, Ferraat, Pascal. They continued in investi- 
gations of their own, and on some points strongly opposed 
Descartes. The universities of France were under strict 
ecclesiastical control and did nothing to introduce his mathe- 
matics and philosophy. It was in the youthful universities of 



DESCARTES TO NEWTON. 189 

Holland that the effect of Cartesian teachings was most 
iiniuediate and strongest. 

The only prominent Frenchman who immediately followed 
in the footsteps of the great master was De Beaune (1601-1652). 
He was one of the first to point out that the properties of a 
curve can be deduced from the properties of its tangent. This 
mode of inquiry has been called the inverse method of tangents. 
He contributed to the theory of equations by considering for 
the first time the upper and lower limits of the roots of 
numerical equations. 

In the Netherlands a large number of distinguished mathema- 
ticians were at once struck with admiration for the Cartesian 
geometry. Foremost among these are van Schooten, John de 
WUt, van Heura£tj SluzCy and Hudde, Van Schooten (died 1660), 
professor of mathematics at Leyden, brought out an edition 
of Descartes' geometry, together with the notes thereon by 
De Beaune. His chief work is his Exercitationes Mathematics, 
in which he applies the analytical geometry to the solution of 
many interesting and difficult problems. The noble-hearted 
Johann de Witt, grand-pensioner of Holland, celebrated as a 
statesman and for his tragical end, was an ardent geometrician. 
He conceived a new and ingenious way of generating conies, 
which is essentially the same as that by projective pencils of 
rays in modern synthetic geometry. He treated the subject 
not synthetically, but with aid of the Cartesian analysis. 
Ren6 Franqois de Sluze (1622-1685) and Johann Hudde (1633- 
1704) made some improvements on Descartes' and Fermat's 
methods of drawing tangents, and on the theory of maxima and 
minima. With Hudde, we find the first use of three variables 
in analytical geometry. He is the author of an ingenious rule 
for finding equal roots. We illustrate it by the equation 
a^ — a^ — 8a;-fl2 = 0. Takinj* an aritlunotioal progression 
3, 2, 1, 0, of which the highest term is equal to the degree of 



190 A HISTORY OF MATHEMATICS. 

the equation, and multiplying each term of the equation respec- 
tively by the corresponding term of the progression, we get 
3aj» — 2aj* — 8a; = 0, or 3aj»-2a;-8 = 0. This last equation 
is by one degree lower than the original one. Find the G.C.D. 
of the two equations. This is a; — 2 ; hence 2 is one of the two 
equal roots. Had there been no common divisor, then the 
original equation would not have possessed equal roots. Hudde 
gave a demonstration for this rule."* 

Heinrich van Heuraet must be mentioned as one of the earli- 
est geometers who occupied themselves with success in the 
rectification of carves. He observed in a general way that the 
two problems of quadrature and of rectification are really 
identical, and that the one can be reduced to the other. Thus 
he carried the rectification of the hyperbola back to the 
quadrature of the hyperbola. The semi-cubical parabola 
2/* = aa? was the first curve that was ever rectified absolutely. 
This appears to have been accomplished independently by Van 
Heuraet in Holland and by William Neil (1637-1670) in Eng- 
land. According to Wallis the priority belongs to Neil. Soon 
after, the cycloid was rectified by Wren and Fermat. 

The prince of philosophers in Holland, and one of the 
greatest scientists of the seventeenth century, was Christian 
Huygens (1629-1695), a native of the Hague. Eminent as a 
physicist and astronomer, as well as mathematician, he was 
a worthy predecessor of Sir Isaac Newton. He studied at 
Ley den under the younger Van Schooten. The perusal of 
some of his earliest theorems led Descartes to predict his 
future greatness. In 1651 Huygens wrote a treatise in which 
he pointed out the fallacies of Gregory St. Vincent (1584-1667) 
on the subject of quadratures. He himself gave a remarkably 
close and convenient approximation to the length of a circular 
arc. In 1660 and 1663 he went to Paris and to London. In 
1666 he was appointed by Louis XIV. member of the French 



DESCARTES TO NEWTON. 191 

Academy of Sciences. He was induced to remain in Paris 
from that time until 1681, when he returned to his native 
city, partly for consideration of his health and partly on 
account of the revocation of the Edict of Nantes. 

The majority of his profound discoveries were made with 
aid of the ancient geometry, though at times he used the 
geometry of Descartes or of Cavalieri and Fermat. Thus, 
like his illustrious friend, Sir Isaac Newton, he always showed 
partiality for the Greek geometry. Newton and Huygens 
were kindred minds, and had the greatest admiration for each 
other. Newton always speaks of him as the ^'Summus 
Hugenius." 

To the two curves (cubical parabola and cycloid) previously 
rectified he added a third, — the cissoid. He solved the 
problem of the catenary, determined the surface of the 
parabolic and hyperbolic conoid, and discovered the proper- 
ties of the logarithmic curve and the solids generated by it. 
Huygens' De horologio oscillator io (Paris, 1673) is a work that 
ranks second only to the Principia of Newton and constitutes 
historically a necessary introduction to it."* The book opens 
with a description of pendulum clocks, of which Huygens 
is the inventor. Then follows a treatment of accelerated 
motion of bodies falling free, or sliding on inclined planes, or 
on given curves, — culminating in the brilliant discovery that 
the cycloid is the tautochronous curve. To the theory of 
curves he added the important theory of "evolutes." After 
explaining that the tangent of the evolute is normal to the 
involute, he applied the theory to the cycloid, and showed by 
simple reasoning that the evolute of this curve is an equal 
cycloid. Then comes the complete general discussion of the 
centre of oscillation. This subject had been proposed for 
investigation by Mersenne and discussed by Descartes and 
Boberval. In Huygens' assumption that the common centre 



192 A HISTORY OF MATHEMATICS. 

of gravity of a group of bodies, oscillating about a horizontal 
axis, rises to its original height, but no higher, is expressed 
for the first time one of the most beautiful principles of 
dynamics, afterwards called the principle of the conservation 
of vis viva.^ The thirteen theorems at the close of the work 
relate to the theory of centrifugal force in circular motion. 
This theory aided Newton in discovering the law of gravita- 
tion. 

Huygens wrote the first formal treatise on probability. He 
proposed the wave-theory of light and with great skill applied 
geometry to its development. This theory was long neglected, 
but was revived and successfully worked out by Young and 
Fresnel a century later. Huygens and his brother improved 
the telescope by devising a better way of grinding and polish- 
ing lenses. With more efficient instruments he determined 
the nature of Saturn's appendage and solved other astro- 
nomical questions. Huygens' OpusctUa posthuma appeared 
in 1703. 

Passing now from Holland to England, we meet there one 
of the most original mathematicians of his day — John Wallis 
(1616-1703). He was educated for the Church at Cambridge 
and entered Holy Orders. But his genius was employed 
chiefly in the study of mathematics. In 1649 he was appointed 
Savilian professor of geometry at Oxford. He was one of 
the original members of the Royal Society, which was founded 
in 1663. Wallis thoroughly grasped the mathematical methods 
both of Cavalieri and Descartes. His Conic Sections is the 
earliest work in which these curves are no longer considered 
as sections of a cone, but as curves of the second degree, and 
are treated analytically by the Cartesiaij method of co-or- 
dinates. In this work Wallis speaks of Descartes in the 
highest terms, but in his Algebra he, without good reason, 
accuses Descartes of plagiarising from Harriot. We have 



DESCABTES TO NEWTON. 193 

already mentioned elsewhere Wallis's solution of the prize 
questions on the cycloid, which were proposed by Pascal. 

The Arithmetic of Infinites, published in 1655, is his greatest 
work. By the application of analysis to the Method of Indi- 
visibles, he greatly increased the power of this instrument for 
effecting quadratures. He advanced beyond Kepler by mak- 
ing more extended use of the " law of continuity " and placing 
full reliance in it. By this law he was led to regard the 
denominators of fractions as powers with negative exponents. 
Thus, the descending geometrical progression as*, jc*, ^, aP, if 
continued, gives x''\ x"^, a?"^, etc. ; which is the same thing 

111 

as -, -1, — . The exponents of this geometric series are in 

X ST QCr 

continued arithmetical progression, 3, 2, 1, 0, — 1, — 2, — 3. 
He also used fractional e:xponents, which, like the negative, 
had been invented long before, but had failed to be generally 
introduced. The symbol oo for infinity is due to him. 

Cavalieri and the French geometers had ascertained the 
formula for squaring the parabola of any degree, y = af^, m 
being a positive integer. By the summation of the powers 
of the terms of infinite arithmetical series, it was found that 
the curve y = «"• is to the area of the parallelogram having 
the same base and altitude as 1 is to m -f 1. Aided by the law 
of continuity, Wallis arrived at the result that this formula 
holds true not only when m is positive and integral, but also 
when it is fractional or negative. Thus, in the parabola 
y = ^lyxy wi = ^ ; hence the area of the parabolic segment 
is to that of the circumscribed rectangle as 1 : 1^, or as 2 : 3. 
Again, suppose that in y = af, m= — ^; then the curve is a 
kind of hyperbola referred to its asymptotes, and the hyper- 
bolic space between the curve and its asymptotes is to the 
corresponding parallelogram as 1 : J. If m= —1, as in the 
common equilateral hyperbola y = x-^ or a?y=l, then this 
ratio is 1 : —1 -f 1, or 1 : 0, showing that its asymptotic space 



194 A HISTORY OF MATHEMATICS. 

is infinite. But in the case when m is greater than unity and 
negative, Wallis was unable to interpret correctly his results. 
For example, if m = — 3, then the ratio becomes 1 : — 2, or 
as unity to a negative number. What is the meaning of this ? 
Wallis reasoned thus : If the denominator is only zero^ then 
the area is already infinite; but if it is less than zero, then the 
area must be more than infinite. It was pointed out later by 
Yarignon, that this space, supposed to exceed infinity, is really 
finite, but taken negatively ; that is, measured in a contrary 
direction.^ The method of Wallis was easily extended to 

m p 

cases such as y = ax* + bxf by performing the quadrature 
for each term separately, and then adding the results. 

The manner in which Wallis studied the quadrature of the 
circle and arrived at his expression for the value of ir is 
extraordinary. He found that the areas comprised between 
the axes, the ordinate corresponding to x, and the curves 
represented by the equations y = (1 — jc')'*, j^ = (1 — aj')^ y=s 
(1 — x*)', y = (1 — a^y, etc., are expressed in functions of the 
circumscribed rectangles having x and y for their sides, by the 
quantities forming the series 

x-^a^y 

a; — |aj^4-f ar* — l^^, etc. 

When aj = l, these values become respectively 1, |, -^, -j^, 
etc. Now since the ordinate of the circle is y = (1 — a:*)*, the 
exponent of which is ^ or the mean value between and 1, the 
question of this quadrature reduced itself to this : If 0, 1, 2, 
3, etc., operated upon by a certain law, give 1, |, -^, y^, 
what will ^ give, when operated upon by the same law ? He 
attempted to solve this by interpoIcUion, a method first brought 
into prominence by him, and arrived by a highly complicated 



DESCARTES TO NEWTON. 195 

and difficult analysis at the following very remarkable ex- 
pression : 

ir 2.2.4.4.6.6.8.8... 



2 1.3.3.5.5.7.7.9... 



He did not succeed in making the interpolation itself, 
because he did not employ literal or general exponents, and 
could not conceive a series with more than one term and less 
than two, which it seemed to him the interpolated series must 
have. The consideration of this difficulty led Newton to the 
discovery of the Binomial Theorem. This is the best place to 
speak of that discovery. Newton virtually assumed that the 
same conditions which underlie the general expressions for 
the areas given above must also hold for the expression to be 
interpolated. In the first place, he observed that in each 
expression the first term is a;, that x increases in odd powers, 
that the signs alternate + and — , and that the second terms 
J x*, I aj*, I a^, I ar*, are in arithmetical progression. Hence the 

first two terms of the interpolated series must be a? — i — . 

He next considered that the denominators 1, 3, 5, 7, etc., are in 
arithmetical progression, and that the coefficients in the nu- 
merators in each expression are the digits of some power of 
the number 11 ; namely, for the first expression, 11® or 1 ; for 
the second, 11^ or 1, 1 ; for the third, 11' or 1, 2, 1 ; for the 
fourth, 11^ or 1, 3, 3, 1 ; etc. He then discovered that, having 
given the second digit (call it m), the remaining digits can 
be found by continual multiplication of the terms of the series 

. — - — . . — — etc. Thus, if m = 4, then 

12 3 4 

4 m — 1 •^«/!/»wi — 2 • jt A tn — 3 . ^ » 
. gives 6 ; o . — - — gives 4 ; 4 • gives 1. Ap- 

2 3 4 

plying this rule to the required series, since the second term 

.„ — , we have m = .V, and then get for the succeeding co- 
«5 



196 A HISTOBY OF MATHEMATICS. 

efficients in the numerators respectively — J, +-^f — y^, 
etc.; hence the required area for the circular segment is 

X — ^ 2_ — IX etc. Thus he found the interpolated 

expression to be an infinite series, instead of one having more 
than one term and less than two, as Wallis believed it must 
be. This interpolation suggested to Newton a mode of ex- 
panding (1 — aj*)*, or, more generally, (1 — aj^)*, into a series. 
He observed that he had only to omit from the expression 
just found the denominators 1, 3, 6, 7, etc., and to lower 
each power of x by unity, and he had the desired expression. 
In a letter to Oldenburg (June 13, 1676), Newton states the 
theorem as follows : The extraction of roots is much shortened 
by the theorem 

(P + PQ)" = P" + 25 ^Q + ^LU* BQ + ^L=2j!i eg + etc., 

n 2n on 

m 

where A means the first term, P», B the second term, C the 
third term, etc. He verified it by actual multiplication, but 
gave no regular proof of it. He gave it for any exponent 
whatever, but made no distinction between the case when the 
exponent is positive and integral, and the others. 

It should here be mentioned that very rude beginnings of 
the binomial theorem are found very early. The Hindoos and 
Arabs used the expansions of (a -f 6)^ and (a -f 6)^ for extract- 
ing roots; Vieta knew the expansion of (a 4-6)*; but these 
were the results of simple multiplication without the discovery 
of any law. The binomial coefficients for positive whole expo- 
nents were known to some Arabic and European mathema- 
ticians. Pascal derived the coefficients from the method of 
what is called the " arithmetical triangle." Lucas de Burgo, 
Stifel, Stevinus, Briggs, and others, all possessed something 
from which one would think the binomial theorem could have 



DESCARTES TO NEWTON. 197 

been gotten with a little attention, " if we did not know that 
such simple relations were difficult to discover." 

Though Wallis had obtained an entirely new expression for 
X, he was not satisfied with it ; for instead of a finite number 
of terms yielding an absolute value, it contained merely an 
infinite number, approaching nearer and nearer to that value. 
He therefore induced his friend, Lord Brouncker (1620 ?-1684), 
the first president of the Royal Society, to investigate this 
subject. Of course Lord Brouncker did not find what they 
were after, but he obtained the following beautiful equality: — 



i-h-^ 



2 + 



2 + -25 



2+ ^9 



2 + etc. 

Continued fractions, both ascending and descending, appear to 
have been known already to the Greeks and Hindoos, though 
not in our present notation. Brouncker's expression gave 
birth to the theory of continued fractions. 

Wallis' method of quadratures was diligently studied by 
his disciples. Lord Brouncker obtained the first infinite 
series for the area of an equilateral hyperbola between its 
asymptotes. Nicolaus Mercator of Holstein, who had settled 
in England, gave, in his Logarithmotechnia (London, 1668), a 
similar series. He started with the grand property of the 
equilateral hyperbola, discovered in 1647 by Gregory St, Vin- 
cent, which connected the hyperbolic space between the 
asymptotes with the natural logarithms and led to these 
logarithms being called hyperbolic. By it Mercator arrived 
at the logarithmic series, which Wallis had attempted but 
failed to obtain. He showed how the construction of logarith- 



198 



A HISTORY OF MATHBMATICS. 



mic tables could be reduced to the quadrature of hyperbolic 
spaces. Following up some suggestions of Wallis, William 
Neil succeeded in rectifying the cubical parabola, and Wren in 
rectifying any cycloidal arc. 

A prominent English mathematician and contemporary of 
Wallis was Isaac Barrow (1630-1677). He was professor of 
mathematics in London, and then in Cambridge, but in 1669 
he resigned his chair to his illustrious pupil, Isaac Newton, 
and renounced the study of mathematics for that of divinity. 
As a mathematician, he is most celebrated for his method of 
tangents. He simplified the method of Fermat by introduc- 
ing two infinitesimals instead of one, and approximated to the 
course of reasoning afterwards followed by Newton in his doc- 
trine on Ultimate Ratios. 

He considered the infinitesimal right triangle ABB' having 
for its sides the difference between two successive ordinates, 
the distance between them, and the portion of the curve inter- 
cepted by them. This triangle is similar to BPT, formed by 
the ordinate, the tangent, and the sub-tangent. Hence, if we 
know the ratio of B'A to BA, then we know the ratio of the 
ordinate and the sub-tangent, and the tangent can be con- 
structed at once. For any curve, say y^ = px, the ratio of B'A 

to BA is determined from its 
equation as follows : If x re- 
ceives an infinitesimal incre- 
ment PP'=ey then y receives 
an increment B'A = a, and the 
equation for the ordinate B'P* 
becom es y^ -f 2 ay -f a* = px -fpe. 
Since y^=zpx, we get 2 ay •{- a^ = pe \ neglecting higher powers 
of the infinitesimals, we have ^ay = pe, which gives 




a:e=2):2y=2^'' ^Vpx. 



NEWTON TO EULBR. 199 

Bnt aiesz the ordinate : the sub-tangent ; hence 

p : 2 Vpi = -y/px : sub-tangent, 

giving 2 X for the value of the sub-tangent. This method dif- 
fers from that of the differential calculus only in notation.^ 



NEWTON TO EULER. 

It has been seen that in France prodigious scientific progress 
was made during the beginning and middle of the seventeenth 
century. The toleration which marked the reign of Henry IV. 
and Louis XIII. was accompanied by intense intellectual 
activity. Extraordinary confidence came to be placed in the 
power of the human mind. The bold intellectual conquests 
of Descartes, Fermat, and Pascal enriched mathematics with 
imperishable treasures. During the early part of the reign 
of Louis XIV. we behold the sunset splendour of this glorious 
period. Then followed a night of mental effeminacy. This 
lack of great scientific thinkers during the reign of Louis XIV. 
may be due to the simple fact that no great minds were born ; 
but, according to Buckle, it was due to the paternalism, to 
the spirit of dependence and subordination, and to the lack 
of toleration, which marked the policy of Louis XIV. 

In the absence of great French thinkers, Louis XIV. sur- 
rounded himself by eminent foreigners. Romer from Den- 
mark, Huygens from Holland, Dominic Cassini from Italy, 
were the mathematicians and astronomers adorning his court. 
They were in possession of a brilliant reputation before going 
to Paris. Simply because they performed scientific work in 
Paris, that work belongs no more to France than the dis- 
coveries of Descartes belong to Holland, or those of Lagrange 
to Germany, or those of Euler and Poncelet to Russia. We 



200 A HISTORY OF MATHEMATICS. 

must look to other countries than France for the great scien- 
tific men of the latter part of the seventeenth century. 

About the time when Louis XIY. assumed the direction 
of the French government Charles II. became king of Eng- 
land. At this time England was extending her commerce 
and navigation^ and advancing considerably in material pros- 
perity. A strong intellectual movement took place, which 
was unwittingly supported by the king. The age of poetry 
was soon followed by an age of science and philosophy. In 
two successive centuries England produced Shakespeare and 
Newton ! 

Germany still continued in a state of national degradation. 
The Thirty Years' War had dismembered the empire and 
brutalised the people. Yet this darkest period of Germany's 
history produced Leibniz, one of the greatest geniuses of 
modem times. 

There are certain focal points in history toward which 
the lines of past progress converge, and from which radiate 
the advances of the future. Such was the age of Newton 
and Leibniz in the history of mathematics. During fifty 
years preceding this era several of the brightest and acutest 
mathematicians bent the force of their genius in a direction 
which finally led to the discovery of the infinitesimal calculus 
by Newton and Leibniz. Cavalieri, Roberval, Fermat, Des- 
cartes, Wallis, and others had each contributed to the new 
geometry. So great was the advance made, and so near 
was their approach toward the invention of the infinitesimal 
analysis, that both Lagrange and Laplace pronounced their 
countryman, Fermat, to be the true inventor of it. The dif- 
ferential calculus, therefore, was not so much an individual 
discovery as the grand result of a succession of discoveries 
by different minds. Indeed, no great discovery ever flashed 
upon the mind at once, and though those of Newton will 



NEWTON TO BULBB. 201 

influence mankind to the end of the world, yet it must be 
admitted that Pope's lines are only a " poetic fancy " : — 

** Nature and Nature^s laws lay hid in night ; 
God said, * Let Newton be/ and all was light'* 

Isaac Kewton (1642-1727) was born at Woolsthorpe, in 
Lincolnshire, the same year in which Galileo died. At his 
birth he was so small and weak that his life was despaired of. 
His mother sent him at an early age to a village school, and 
in his twelfth year to the public school at Grantham. At 
first he seems to have been very inattentive to his studies 
and very low in the school; but when, one day, the little 
Isaac received a severe kick upon his stomach from a boy 
who was above him, he laboured hard till he ranked higher 
in school than his antagonist. From that time he continued 
to rise until he was the head boy.® At Grantham, Isaac 
showed a decided taste for mechanical inventions. He con- 
structed a water-clock, a wind-mill, a carriage moved by the 
person who sat in it, and other toys. When he had attained 
his fifteenth year his mother took him home to assist her in 
the management of the farm, but his great dislike for farm- 
work and his irresistible passion for study, induced her to 
send him back to Grantham, where he remained till his 
eighteenth year, when he entered Trinity College, Cambridge 
(1660). Cambridge was the real birthplace of Newton's 
genius. Some idea of his strong intuitive powers may be 
drawn from the fact that he regarded the theorems of ancient 
geometry as self-evident truths, and that, without any prelimi- 
nary study, he made himself master of Descartes' Geometry. 
He afterwards regarded this neglect of elementary geometry 
a mistake in his mathematical studies, and he expressed to 
Dr. Pemberton his regret that " he had applied himself to the 
works of Descartes and other algebraic writers before he had 



202 A HISTORY OF MATHEMATICS. 

considered the Elements of Euclid with that attention which 
so excellent a writer deserves," Besides Descartes' Chometry, 
he studied Oughtred's Clavis, Kepler's OpticSj the works of 
Vieta, Schooten's Miscellanies, Barrow's Lectures, and the 
works of Wallis. He was particularly delighted with Wallis' 
Arithmetic of Infinites, a treatise fraught with rich and varied 
suggestions. Newton had the good fortune of having for 
a teacher and fast friend the celebrated Dr. Barrow, who 
had been elected professor of Greek in 1660, and was made 
Lucasian professor of mathematics in 1663. The mathe- 
matics of Barrow and of Wallis were the starting-points 
from which Newton, with a higher power than his masters', 
moved onward into wider fields. Wallis had effected the 
quadrature of curves whose ordinates are expressed by any 
integral and positive power of (1 — «*). We have seen how 
Wallis attempted but failed to interpolate between the areas 
thus calculated, the areas of other curves, such as that of 
the circle; how Newton attacked the problem, effected the 
interpolation, and discovered the Binomial Theorem, which 
afforded a much easier and direct access to the quadrature 
of curves than did the method of interpolation ; for even 
though the binomial expression for the ordinate be raised 
to a fractional or negative power, the binomial could at once 
be expanded into a series, and the quadrature of each separate 
term of that series could be effected by the method of Wallis. 
Newton introduced the system of literal indices. 

Newton's study of quadratures soon led him to another 
and most profound invention. He himself says that in 1665 
and 1666 he conceived the method of fluxions and applied 
them to the quadrature of curves. Newton did not com- 
municate the invention to any of his friends till 1669, when 
he placed in the hands of Barrow a tract, entitled De Analysi 
per ^quationes Numero Terminorum Injinitas, which was sent 



NEWTON TO EULBR. 203 

by Barrow to Collins, who greatly admired it. In this treatise 
the principle of fluxions, though distinctly pointed out, is only 
partially developed and explained. Supposing the abscissa 
to increase uniformly in proportion to the time, he looked 
upon the area of a curve as a nascent quantity increasing 
by continued fluxion in the proportion of the length of the 
ordinate. The expression which was obtained for the fluxion 
he expanded into a finite or infinite series of monomial terms, 
to which Wallis' rule was applicable. Barrow urged Newton 
to publish this treatise ; " but the modesty of the author, of 
which the excess, if not culpable, was certainly in the present 
instance very unfortunate, prevented his compliance." * Had 
this tract been published then, instead of forty-two years 
later, there would probably have been no occasion for that 
long and deplorable controversy between Newton and Leibniz. 

For a long time Newton's method remained unknown, ex- 
cept to his friends and their corresix)ndent8. In a letter 
to Collins, dated December 10th, 1672, Newton states the fact 
of his invention with one example, and then says: "This 
is one particular, or rather corollary, of a general method, 
which extends itself, without any troublesome calculation, not 
only to the drawing of tangents to any curve lines, whether 
geometrical or mechanical, or anyhow respecting right lines 
or other curves, but also to the resolving other abstruseijf 
kinds of problems about the crookedness, areas, lengths! 
centres of gravity of curves, etc.; nor is it (as Hudden'j 
method of Maximis and Minimis) limited to equations whicl^ 
are free from surd quantities. This method I have inter\ 
woven with that other of working in equations, by reducingj 
them to infinite series." 

These last words relate to a treatise he composed in the 
year 1671, entitled Method of Fluxions, in which he aimed 
to represent his method as an independent calculus and as 



204 A HISTORY OF MATHEMATICS. 

a complete system. This tract was intended as an introduc- 
tion to an edition of Kinckhuysen's Algebray which he had 
undertaken to publish. " But the fear of being involved in 
disputes about this new discovery, or perhaps the wish to 
render it more complete, or to have the sole advantage of 
employing it in his physical researches, induced him to aban- 
don this design." " 

Excepting two papers on optics, all of his works appear 
to have been published only after the most pressing solicita- 
tions of his friends and against his own wishes.^ His re- 
searches on light were severely criticised, and he wrote in 
1675: "I was so persecuted with discussions arising out of 
my theory of light that I blamed my own imprudence for 
parting with so substantial a blessing as my quiet to run 
after a shadow." 

The Method of Fluxions, translated by J. Colson from New- 
ton's Latin, was first published in 1736, or sixty-five years 
after it was written. In it he explains first the expansion 
into series of fractional and irrational quantities, — a subject 
which, in his first years of study, received the most careful 
attention. He then proceeds to the solution of the two fol- 
lowing mechanical problems, which constitute the pillars, so 
to speak, of the abstract calculus : — 

"I. The length of the space described being continually 
(i.e. at all times) given ; to find the velocity of the motion at 
any time proposed. 

" II. The velocity of the motion being continually given ; to 
find the length of the space described at any time proposed." 

Preparatory to the solution, Newton says: "Thus, in the 
equation y = s^, if y represents the length of the space at any 
time described, which (time) another space a;, by increasing 
with an uniform celerity jc, measures and exhibits as described : 
then 2 XX will represent the celerity by which the space y, 



NEWTON TO EULER. 205 

at the same moment of time, proceeds to be described; and 
contrarywise." 

" But whereas we need not consider the time here, any far- 
ther than it is expounded and measured by an equable local 
motion; and besides, whereas only quantities of the same 
kind can be compared together, and also their velocities of 
increase and decrease ; therefore, in what follows I shall have 
no regard to time formally considered, but I shall suppose 
some one of the quantities proposed, being of the same kind, 
to be increased by an equable fluxion, to which the rest may be 
referred, as it were to time ; and, therefore, by way of analogy, 
it may not improperly receive the name of time." In this 
statement of Newton there is contained a satisfactory answer 
to the objection which has been raised against his method, 
that it introduces into analysis the foreign idea of motion. A 
quantity thus increasing by uniform fluxion, is what we now 
call an independent variable. 

Newton continues : " Now those quantities which I consider 
as gradually and indefinitely increasing, I shall hereafter call 
fluents, or flowing quantities, and shall represent them by the 
final letters of the alphabet, v, x, y, and 2 ; . . . and the veloci- 
ties by which every fluent is increased by its generating motion 
(which I may call fluxions, or simply velocities, or celerities), 
I shall represent by the same letters pointed, thus, v, x, y, z. 
That is, for the celerity of the quantity i; I shall put v, and so 
for the celerities of the other quantities x, y, and z, I shall put 
X, y, and z, respectively." It must here be observed that New- 
ton does not take the fluxions themselves infinitely small. 
The " moments of fluxions," a term introduced further on, are 
infinitely small quantities. These " moments," as defined and 
used in the Method of Fluxions, are substantially the differen- 
tials of Leibniz. De Morgan points out that no small amount of 
confusion has arisen from the use of the word fluxion and the 



206 A HISTORY OF MATHEMATICS. 

notation £ by all the English writers previous to 1704^ except- 
ing Newton and Cheyne^ in the sense of an infinitely small in- 
crement." Strange to say, even in the Commercium Epistolicum 
the words momeftU dji^ fluent appear to be used as synonymous. 

After showing by examples how to solve the first problem, 
Newton proceeds to the demonstration of his solution : — 

''The moments of flowing quantities (that is, their indefi- 
nitely small parts, by the accession of which, in infinitely 
small portions of time, they are continually increased) are as 
the velocities of their flowing or increasing. 

" Wherefore, if the moment of any one (as x) be represented 
by the product of its celerity x into an infinitely small quantity 
(i.e. by AO), the moments of the others, v, y, z, will be repre- 
sented by vO, yO, iO; because tK), aO, jfO, and 2O are to each other 
as V, X, py and z. 

" Now since the moments, as xO and yO, are the indefinitely 
little accessions of the flowing quantities x and y, by which 
those quantities are increased through the several indefinitely 
little intervals of time, it follows that those quantities, x 
and y, after any indefinitely small interval of time, become 
a? 4- xO and y 4- yO, and therefore the equation, which at all 
times indifferently expresses the relation of the flowing quan- 
tities, will as well express the relation between x-\- xO and 
y 4- yO, as between x and y ; so that x-\-xO and y + yO may 
be substituted in the same equation for those quantities, in- 
stead of X and y. Thus let any equation a^ — aa^ + axy — y^ = 
be given, and substitute x + iO for a?, and y -F yO for y, and 
there will arise 

a? 4-3aj2A0 -{-SxxOxO-^'d^ 
— aaP — 2axiO — axOxO 
-f axy -f- ayxO + axOyO V = 0. 

■haxyO I 

-f -3y*y0 -3yy0y0-yWj 



NEWTON TO EULBB, 207 

" Now, by supposition, a^ — oa^ -f axy — y* = 0, which there- 
fore, being expunged and the remaining terms being divided 
V>y 0, there will remain 

3a^x — 2axx -f ayx -f- axy — Sy^y 4- 3xa»0 — axxO -f- axyO 

- '^yyyo + i^o - y^o = o. 

IBut whereas zero is supposed to be infinitely little, that it 
Uay represent the moments of quantities, the terms that are 
xiultiplied by it will be nothing in respect of the rest (termini 
in earn ducii pro nihilo possunt haberi cum cUiis coUati) ; there- 
fore I reject them, and there remains 

Sa^x — 2aa;i 4- «y^ 4- cixy — ^y^y = 0, 

as above in Example I." Newton here uses infinitesimals. 

Much greater than in the first problem were the difficulties 
encountered in the solution of the second problem, involving, 
as it does, inverse operations which have been taxing the skill 
of the best analysts since his time. Newton gives first a 
special solution to the second problem in which he resorts 
to a rule for which he has given no proof. 

In the general solution of his second problem, Newton 
assumed homogeneity with respect to the fluxions and then 
considered three cases: (1) when the equation contains two 
fluxions of quantities and but one of the fluents; (2) when 
the equation involves both the fluents as well as both the flux- 
ions ; (3) when the equation contains the fluents and the flux- 
ions of three or more quantities. The first case is the easiest 

since it requires simply the integration of -2 =/(»), to which 

dx 

his "special solution" is applicable. The second case de- 
manded nothing less than the general solution of a dif- 
ferential equation of the first order. Those who know what 
efforts were afterwards needed for the complete exploration 
of this field in analysis, will not depreciate Newton's work 



208 A UISTOBY OF MATHBMATIGS. 

even though he resorted to solutions in form of infinite series. 
Newton's third case comes now under the solution of partial 
differential equations. He took the equation 2a— i;4-ficy = 
and succeeded in finding a particular integral of it. 

The rest of the treatise is devoted to the determination of 
maxima and minima, the radius of curvature of curves, and 
other geometrical applications of his fluxionary calculus. All 
this was done previous to the year 1672. 
[1 It must be observed that in the Method of Fluxions (as well 
as in his De Analyst and all earlier papers) the method 
employed by Newton is strictly infinitesimal, and in substance 
like that of Leibniz. Thus, the original conception of the 
calculus in England, as well as on the Continent, was based 
on infinitesimals. The fundamental principles of the fluxionary 
calculus were first given to the world in the Principia ; but its 
peculiar notation did not appear until published in the second 
volume of Wallis' Algebra in 1693. The exposition given in 
the Algebra was substantially a contribution of Newton ; it 
rests on infinitesimals. In the first edition of the Principia 
(1687) the description of fluxions is likewise founded on 
infinitesimals, but in the second (1713) the foundation is 
somewhat altered. In Book II. Lemma II. of the first edition 
we read : " Cave tarn en intellexeris particulas finitas. Momenta 
quam primum finitce sunt ma^nitudinis, desinunt esse momenta. 
Finiri enim repitgnat allquatenus perpetuo eorum incremento vet 
decremento. Intelligenda sunt principia jamjam nascentia 
finitorum magnitudinum." In the second edition the two 
sentences which we print in italics are replaced by the 
following : " Particulse finitae non sunt momenta sed quanti- 
tates ipsae ex momentis genitse." Through the difficulty of 
the phrases in both extracts, this much distinctly appears, 
that in the first, moments are infinitely small quantities. 
What else they are in the second is not clear.*^ In the 



NEWTON TO EULBR. 209 

Quadrature of Curves of 1704, the infinitely small quantity is 
completely abandoned. It has been shown that in the Method 
of Fluxions Newton rejected terms involving the quantity 0, 
because they are infinitely small compared with other terms. 
This reasoning is evidently erroneous; for as long as is a 
quantity, though ever so small, this rejection cannot be made 
without affecting the result. Newton seems to have felt this, 
for in the Quadrature of Curves he remarked that " in math- 
ematics the minutest errors are not to be neglected " (errores 
quam minimi in rebus mathematicis non sunt contemnendi). 

The early distinction between the system of Newton and 
Leibniz lies in this, that Newton, holding to the conception \ 
of velocity or fluxion, used the infinitely small increment as a 
means of determining it, while with Leibniz the relation of the 
infinitely small increments is itself the object of determination. 
The difference between the two rests mainly upon a difference 
in the mode of generating quantities." 

We give Newton's statement of the method of fluxions or 
rates, as given in the introduction to his Quadrature of Curves, 
"I consider mathematical quantities in this place not as 
consisting of very small parts, but as described by a continued 
motion. Lines are described, and thereby generated, not by 
the apposition of parts, but by the continued motion of points ; 
superficies by the motion of lines; solids by the motion of 
superficies ; angles by the rotation of the sides ; portions of 
time by continual flux : and so on in other quantities. These 
geneses really take place in the nature of things, and are 
daily seen in the motion of bodies. . . . 

" Fluxions are, as near as we please (quam proxime), as the 
increments of fluents generated in times, equal and as small as 
possible, and to speak accurately, they are in the prime ratio 
of nascent increments ; yet they can be expressed by any lines 
whatever, which are proportional to them." 



210 



A HISTORY OF MATHEMATICS. 



Newton exemplifies this last assertion by the problem of 
tangency : Let AB be the abscissa^ BC the ordinate, VCH 
the tangent, Ec the increment of the ordinate, which pro- 
duced meets FIT at T, and Cc the increment of the curve. 
The right line Cc being produced to Kj there are formed 
three small triangles^ the rectilinear CEc^ the mixtilinear 
CEc, and the rectilinear CET, Of these, the first is evidently 
the smallest, and the last the greatest. Now suppose 
the ordinate be to move into the place BCj so that the 

point c exactly co- 
incides with the 
point (7; CKj and 
therefore the curve 
Cc, is coincident 
with the tangent 
CH, Ec is abso- 
lutely equal to ET, 
and the mixtilinear 
evanescent triangle CEc is, in the last form, similar to the 
triangle CET; and its evanescent sides CE, Ec, Cc, will be 
proportional to CE, ET, and CT, the sides of the triangle 
CET. Hence it follows that the fluxions of the lines AB, 
BCj AC, being in the last ratio of their evanescent increments, 
are proportional to the sides of the triangle CET, or, which is 
all one, of the triangle VBC similar thereunto. As long as 
the points C and c are distant from each other by an interval, 
however small, the line CK will stand apart by a small angle 
from the tangent CH, But when CK coincides with CH, and 
the lines CE, Ec, cC reach their ultimate ratios, then the 
points C and c accurately coincide and are one and the same. 
Newton then adds that " in mathematics the minutest errors 
are not to be neglected." This is plainly a rejection of the 
postulates of Leibniz. The doctrine of infinitely small quan- 




NEWTON TO BULBR. 211 

titles is here renounced in a manner which would lead one 
to suppose that Newton had never held it himself. Thus it 
appears that Newton's doctrine was different in different 
periods. Though, in the above reasoning, the Gharybdis of 
infinitesimals is safely avoided, the dangers of a Scylla stare 
us in the face. We are required to believe that a point may 
be considered a triangle, or that a triangle can be inscribed 
in a point ; nay, that three dissimilar triangles become similar 
and equal when they have reached their ultimate form in one 
and the same point. 

In the introduction to the QuadrcUure of Curves the fluxion 
of a5* is determined as follows : — 

" In the same time that a;, by flowing, becomes a; + 0, the 
power af becomes (a?-f-0)", i.e. by the method of inflnite series 

af 4- nO a;*-^ + ^^^^-^^O^ af-« + etc., 

and the increments 

and nO a;" ^ 4- ^^-p^O'x*"* + etc., 

are to one another as 

1 to nar-^ 4- ^^^0 af-« 4- etc. 



"Let now the increments vanish, and their last proportion 
will be 1 to nx*~M hence the fluxion of the quantity a; is to 
the fluxion of the quantity af* as 1 : naf'^, 

" The fluxion of lines, straight or curved, in all cases what- 
ever, as also the fluxions of superficies, angles, and other 
quantities, can be obtained in the same manner by the method 
of prime and ultimate ratios. But to establish in this way 
the analysis of infinite quantities, and to investigate prime 
and ultimate ratios of finite quantities, nascent or evanescent, 
is in harmony with the geometry of the ancients ; and I have 
endeavoured to show that, in the method of fluxions, it is not 



212 A HISTOBY OF MATHEBIATICS. 

necessary to introduce into geometry infinitely small quanti- 
ties.'' This mode of differentiating does not remove all the 
difficulties connected with the subject. When becomes 



nothing, then we get the ratio - = naf^\ which needs further 

elucidation. Indeed, the method of Newton, as delivered by 
himself, is encumbered with difficulties and objections. Among 
the ablest admirers of Newton, there have been obstinate dis- 
putes respecting his explanation of his method of '^ prime and 
ultimate ratios.'* 

The so-called " method of limits " is frequently attributed 
to Newton, but the pure method of limits was never adopted 
by him as his method of constructing the calculus. All he 
did was to establish in his Prindpia certain principles which 
are applicable to that method, but which he used for a different 
purpose. The first lemma of the first book has been made the 
foundation of the method of limits : — 

" Quantities and the ratios of quantities, which in any finite 
time converge continually to equality, and before the end of 
that time approach nearer the one to the other than by any 
given difference, become ultimately equal." 

In this, as well as in the lemmas following this, there are 
obscurities and difficulties. Newton appears to teach that a 
variable quantity and its limit will ultimately coincide and be 
equal. But it is now generally agreed that in the clearest 
statements which have been made of the theory of limits, the 
variable does not actually reach its limit, though the variable 
may approach it as near as we please. 

The full title of Newton's Principia is Philosophic^ Natura- 
lis Principia Mathematica. It was printed in 1687 under 
the direction, and at the expense, of Dr. Edmund Halley. 
A second edition was brought out in 1713 with many altera- 
tions and improvements, and accompanied by a preface from 



NSWTON TO BULBB. 218 

Mr. Cotes. It was sold out in a few months, but a pirated 
edition published in Amsterdam supplied the demand.^ The 
third and last edition which appeared in England during 
Newton's lifetime was published in 1726 by Henry Pemberton. 
The Principia consists of three books, of which the first two, 
constituting the great bulk of the work, treat of the mathe- 
matical principles of natural philosophy, namely, the laws and 
conditions of motions and forces. In the third book is drawn 
up the constitution of the universe as deduced from the fore- 
going principles. The great principle underlying this memor- 
able work is that of universal gravitation. The first book 
was completed on April 28, 1686. After the remarkably short 
period of three months, the second book was finished. The 
third book is the result of the next nine or ten months' 
labours. It is only a sketch of a much more extended elabora- 
tion of the subject which he had planned, but which was never 
brought to completion. 

The law of gravitation is enunciated in the first book. Its 
discovery envelops the name of Newton in a halo of perpetual 
glory. The current version of the discovery is as follows : it 
was conjectured by Hooke, Huygens, Halley, Wren, Newton, 
and others, that, if Kepler's third law was true (its absolute 
accuracy was doubted at that time), then the attraction 
between the earth and other members of the solar system 
varied inversely as the square of the distance. But the proof 
of the truth or falsity of the guess was wanting. In 1666 
Newton reasoned, in substance, that if g represent the acceler- 
ation of gravity on the surface of the earth, r be the earth's 
radius, R the distance of the moon from the earth, T the time 
of lunar revolution, and a a degree at the equator, then, if the 
law is true. 



214 A HISTORY OF MATHEMATICS. 

The data at Newton's command gave B = 60.4 r, T = 2,360,628 
seconds, but a only 60 instead of 69} English miles. This 
wrong value of a rendered the calculated value of g smaller 
than its true value, as known from actual measurement. It 
looked as though the law of inverse squares were not the true 
law, and Newton laid the calculation aside. In 1684 he casu- 
ally ascertained at a meeting of the Boyal Society that Jean 
Picard had measui'ed an arc of the meridian, and obtained a 
more accurate value for the earth's radius. Taking the cor- 
rected value for a, he found a figure for g which corresponded 
to the known value. Thus the law of inverse squares was 
verified. In a scholium in the PHncipia, Newton acknowl- 
edged his indebtedness to Huygens for the laws on centrifugal 
force employed in his calculation. 

The perusal by the astronomer Adams of a great mass of 
unpublished letters and manuscripts of Newton forming the 
Portsmouth collection (which remained private property 
until 1872, when its owner placed it in the hands of the 
University of Cambridge) seems to indicate that the difficul- 
ties encountered by Newton in the above calculation were of a 
different nature. According to Adams, Newton's numerical 
verification was fairly complete in 1666, but Newton had not 
been able to determine what the attraction of a spherical shell 
upon an external point would be. His letters to Halley show 
that he did not suppose the earth to attract as though all its 
mass were concentrated into a point at the centre. He could 
not have asserted, therefore, that the assumed law of gravity 
was verified by the figures, though for long distances he might 
have claimed that it yielded close approximations. When 
Halley visited Newton in 1684, he requested Newton to deter- 
mine what the orbit of a planet would be if the law of attrac- 
tion were that of inverse squares. Newton had solved a 
similar problem for Hooke in 1679, and replied at once that it 



NEWTON TO EULBR. 216 

was an ellipse. After Halley's visit, Newton, with Picard's 
new value for the earth's radius, reviewed his early calculsr 
tion, and was able to show that if the distances between the 
bodies in the solar system were so great that the bodies might 
be considered as points, then their motions were in accordance 
vrith the assumed law of gravitation. In 1685 he completed 
his discovery by showing that a sphere whose density at any 
point depends only on the distance from the centre attracts 
an external point as though its whole mass were concentrated 
at the centre.** 

Newton's unpublished manuscripts in the Portsmouth col- 
lection show that he had worked out, by means of fluxions and 
fluents, his lunar calculations to a higher degree of approxima- 
tion than that given in the Principia, but that he was unable 
to interpret his results geometrically. The papers in that col- 
lection throw light upon the mode by which Newton arrived 
at some of the results in the Principia, as, for instance, the 
famous construction in Book II., Prop. 25, which is unproved 
in the Principia, but is demonstrated by him twice in a draft 
of a letter to David Gregory, of Oxford.** 

It is chiefly upon the Principia that the fame of Newton 
rests. Brewster calls it " the brightest page in the records of 
human reason." Let us listen, for a moment, to the comments 
of Laplace, the foremost among those followers of Newton who 
grappled with the subtle problems of the motions of planets 
under the influence of gravitation: "Newton has well estab- 
lished the existence of the principle which he had the merit 
of discovering, but the development of its consequences and 
advantages has been the work of the successors of this great 
mathematician. The imperfection of the infinitesimal calcu- 
lus, when first discovered, did not allow him completely to 
resolve the difficult problems which the theory of the universe 
offers ; and he was oftentimes forced to give mere hints, which 



216 A HISTORY OF MATHEMATICS. 

were always uncertain till confirmed by rigorous analysis. 
Notwithstanding these unavoidable defects, the importance 
and the generality of his discoveries respecting the system of 
the universe, and the most interesting points of natural phi- 
losophy, the great number of profoiind and original views, 
which have been the origin of the most brilliant discoveries of 
the mathematicians of the last century, which were all pre- 
sented with much elegance, will insure to the Principia a last- 
ing pre-eminence over all other productions of the human 
mind." 

Newton's Arithmetica UniversaliSy consisting of algebraical 
lectures delivered by him during the first nine years he was 
professor at Cambridge, were published in 1707, or more than 
thirty years after they were written. This work was pub- 
lished by Mr. Whiston. We are not accurately informed how 
Mr. Whiston came in possession of it, but according to some 
authorities its publication was a breach of confidence on his 
part. 

The Arithmetica UniverscUis contains new and important 
results on the theory of equations. His theorem on the 
sums of powers of roots is well known. Newton showed 
that in equations with real coefficients, imaginary roots always 
occur in pairs. His inventive genius is grandly displayed 
in his rule for determining the inferior limit of the number 
of imaginary roots, and the superior limits for the number 
of positive and negative roots. Though less expeditious than 
Descartes', Newton's nile always gives as close, and generally 
closer, limits to the number of positive and negative roots. 
Newton did not prove his rule. It awaited demonstration 
for a century and a half, until, at last, Sylvester established 
a remarkable general theorem which includes Newton's rule 
as a special case. 

The treatise on Method ofFlitxions contains Newton's method 



NEWTON TO EULER* 217 

of approximating to the roots of numerical equations. This 
is simply the method of Vieta improved. The same treatise 
contains " Newton's parallelogram," which enabled him, in an 
equation, /(oj, y) = 0, to find a series in powers of x equal to 
"the variable y. The great utility of this rule lay in its deter- 
mining the form of the series ; for, as soon as the law was 
luiown by which the exponents in the series vary, then the 
expansion could be effected by the method of indeterminate 
coefficients. The nile is still used in determining the infinite 
branches to curves, or their figure at multiple points. Newton 
gave no proof for it, nor any clue as to how he discovered it. 
The proof was supplied half a century later, by Kaestner and 
Cramer, independently.'' 

In 1704 was published, as an appendix to the Opticks, the 
Bnumeratio Unearum tertii ordinis, which contains theorems 
on the theory of curves. Newton divides cubics into seventy- 
two species, arranged in larger groups, for which his com- 
mentators have supplied the names " genera " and " classes," 
^recognising fourteen of the former and seven (or four) of the 
latter. He overlooked six species demanded by his principles 
of classification, and afterwards added by Stirling, Murdoch, 
and Cramer. He enunciates the remarkable theorem that the 
five species which he names "divergent parabolas" give by 
their projection every cubic curve whatever. As a rule, the 
tract contains no proofs. It has been the subject of frequent 
conjecture how Newton deduced his results. Recently we have 
gotten at the facts, since much of the analysis used by Newton 
and a few additional theorems have been discovered among the 
Portsmouth papers. An account of the four holograph man- 
uscripts on this subject has been published by W. W. Rouse 
Ball, in the Transactions of the London Mathematical Society 
(vol. XX., pp. 104-143). It is interesting to observe how 
Newton begins his research on the classification of cubic 



218 A HISTORY OF MATHEMATICS. 

curves by the algebraic method, but, finding it laborious, 
attacks the problem geometrically, and afterwards returns 
again to analysis." 

Space does not permit us to do more than merely mention 
Newton's prolonged researches in other departments of science. 
He conducted a long series of experiments in optics and is the 
author of the corpuscular theory of light. The last of a 
number of papers on optics, which he contributed to the Eoyal 
Society, 1687, elaborates the theory of "fits." He explained 
the decomposition of light and the theory of the rainbow. By 

■ 

him were invented the reflecting telescope and the sextant 
(afterwards re-discovered by Thomas Godfrey of Philadelphia' 
and by John Hadley), He deduced a theoretical expression 
for the velocity of sound in air, engaged in experiments on 
chemistry, elasticity, magnetism, and the law of cooling, and 
entered upon geological speculations. 

During the two years following the close of 1692, Newton 
suffered from insomnia and nervous irritability. Some thought 
that he laboured under temporary mental aberration. Though 
he recovered his tranquillity and strength of mind, the time 
of great discoveries was over; he would study out questions 
propounded to him, but no longer did he by his own accord 
enter upon new fields of research. The most noted investi- 
gation after his sickness was the testing of his lunar theory 
by the observations of Mamsteed, the astronomer royal. In 
1695 he was appointed warden, and in 1699 master, of the 
mint, which office he held until his death. His body was 
interred in Westminster Abbey, where in 1731 a magnificent 
monument was erected, bearing an inscription ending with, 
" Sibi gratulentur mortales i^ale tantumque exstitisse humani 
generis decus." It is not true that the Binomial Theorem is 
also engraved on it. 

We pass to Leibniz, the second and independent inventor 



NEWTON TO EULEB. 219 

of the calculus. Gottfried Wilhelm Leibniz (1646-1716) was 
bom in Leipzig. No period in the history of any civilised 
nation could have been less favourable for literary and scientific 
pursuits than the middle of the seventeenth century in Ger- 
xnany. Yet circumstances seem to have happily combined 
\o bestow on the youthful genius an education hardly other- 
^wise obtainable during this darkest period of German history. 
Se was brought early in contact with the best of the culture 
then existing. In his fifteenth year he entered the University 
of Leipzig. Though law was his principal study^ he applied 
himself with great diligence to every branch of knowledge. 
Instruction in German universities was then very low. The 
higher mathematics was not taught at all. We are told that 
a certain Johil Kuhn lectured on Euclid's Eleinents, but that 
his lectures were so obscure that none except Leibniz could 
understand them. Later on, Leibniz attended, for a half-year, 
at Jena, the lectures of Erhard Weigel, a philosopher and 
mathematician of local reputation. In 1666 Leibniz published 
a treatise, De Arte Combinatoriay in which he does not pass 
beyond the rudiments of mathematics. Other theses written 
by him at this time were metaphysical and juristical in 
character. A fortunate circumstance led Leibniz abroad. In 
1672 he was sent by Baron Boineburg on a political mission 
to Paris. He there formed the acquaintance of the most 
distinguished men of the age. Among these was Huygens, 
who presented a copy of his work on the oscillation of the 
pendulum to Leibniz, and first led the gifted young German 
to the study of higher mathematics. In 1673 Leibniz went 
to London, and remained there from January till March. He 
there became incidentally acquaintec^ with the mathematician 
Pell, to whom he explained a method he had found on the 
summation of series of numbers by their differences. Pell 
told him that a similar formula had been published by Mouton 



220 A HISTORY OF MATHEMATIGS. 

as early as 1670, and then called his attention to Mercator's 
work on the rectification of the parabola. While in London, 
Leibniz exhibited to the Boyal Society his arithmetical ma- 
chine, which was similar to Pascal's, but more efficient and 
perfect. After his return to Paris, he had the leisure to study 
mathematics more systematically. With indomitable energy 
he set about removing his ignorance of higher mathematics. 
Huygens was his principal master. He studied the geometric 
works of Descartes, Honorarius Fabri, Gregory St. Vincent, 
and Pascal. A careful study of infinite series led him to the 
discovery of the following expression for the ratio of the 
circumference to the diameter of the circle : — 

j = l-i + i-| + i-etc. 

This elegant series was found in the same way as Mercator's 
on the hyperbola. Huygens was highly pleased with it and 
urged him on to new investigations. Leibniz entered into a 
detailed study of the quadrature of curves and thereby became 
intimately acquainted with the higher mathematics. Among 
the papers of Leibniz is still found a manuscript on quadra- 
tures, written before he left Paris in 1676, but which was 
never printed by him. The more important parts of it were 
embodied in articles published later in the Acta Ervditorum. 

In the study of Cartesian geometry the attention of Leibniz 
was drawn early to the direct and inverse problems of tan- 
gents. The direct problem had been solved by Descartes for 
the simplest curves only; while the inverse had completely 
transcended the power of his analysis. Leibniz investigated 
both problems for any curve ; he constructed what he called 
the triangtUum characteristicum — an infinitely small triangle 
between the infinitely small part of the curve coinciding with 
the tangent, and the differences of the ordinates and abscissas. 



NEWTON TO ETJLEB. 221 

A c33C3H:e4s.tere or Mna\^ ^ jf> ^ {\ in inm w |iii1ynnp The tviangulum 
iaracterisUcum is similax to the triangle formed by the tan- 
gent, the ordinate of the point of contact, and the sub- 
tangent, as well as to that between the ordinate, normal, 
and sub-normal. It was first employed by Barrow in Eng- 
land, but appears to have been reinvented by Leibniz. From 
it Leibniz observed the connection existing between the direct' 
9iid inverse problems of tangents. He saw also that the latter , 
csoold be carried back to the quadrature of curves. All these ; 
results are contained in a manuscript of Leibniz, written in! 
^673. One mode used by him in effecting quadratures was 
«8 follows: The rectangle formed by a sub-tangent p and 
an element a (t.e. infinitely small part of the abscissa) is 
equal to the rectangle formed by the ordinate y and the ele- 
ment I of that ordinate; or in symbols, pa^yl. But the 
summation of these rectangles from zero on gives a right 
triangle equal to half the square of the ordinate. Thus, 
using Cavalieri's notation, he gets 

omn. pa = omn. yl=^^ {pmn. meaning omnia^ all). 
But ^=5 omn. 2; hence 



I omn. P 



omn. omn. / - = 

a 2a 

This equation is especially interesting, since it is here that 
Leibniz first introduces a new notation. He says: '^It will 
be useful to write i for omn,, as t / for omn, I, that is, the 
sum of the Ts " ; he then writes the equation thus : — 

2o J J a 
From this he deduced the simplest integrals, such as 

a? 



Jx^-, JC^ + y) =/« + ]■». 



222 A HISTORY OF MATHEMATICS. 

Since the symbol of summation J raises the dimensions, he 
concluded that the opposite calculus, or that of differences 

d, would lower them. Thus, if \l = ya, then l=:^» The 

symbol d was at first placed by Leibniz in the denominator, 
because the lowering of the power of a term was brought 
about in ordinary calculation by division. The manuscript 
giving the above is dated October 29th, 1675.* This, then, 
was the memorable day on which the notation of the new 
calculus came to be, — a notation which contributed enor- 
mously to the rapid growth and perfect development of the 
calculus. 

Leibniz proceeded to apply his new calculus to the solution 
of certain problems then grouped together under the name 
of the Inverse Problems of Tangents. He found the cubical 
parabola to be the solution to the following: To find the 
curve in which the sub-normal is reciprocally proportional 
to the ordinate. The correctness of his solution was tested 
by him by applying to the result Sluze's method of tangents 
and reasoning backwards to the original supposition. In the 
solution of the third problem he changes his notation from 

- to the now usual notation da:. It is worthy of remark 
d 

that in these investigations, Leibniz nowhere explains the 
significance of dx and dy, except at one place in a marginal 

note: "Idem est dx et -, id est, differentia inter duas x 

d' ' 

proximas." Nor does he use the term differential, but 
always difference. Not till ten years later, in the Acta 
Eruditorum, did he give further explanations of these sym- 
bols. What he aimed at principally was to determine the 
change an expression undergoes when the symbol J or d is 
placed before it. It may be a consolation to students wres- 
tling with the elements of the differential calculus to know 
that it reqmred Leibniz considerable thought and atten- 



NEWTON TO EULBB. 228 

tion** to determine whether dxdy is the same as d{icy), and 
- — the same as d-. After considering these questions at 

<iy y 

"the close of one of his manuscripts, he concluded that the 
expressions were not the same, though he could not give the 
true value for each. Ten days later, in a manuscript dated 
Kovember 21, 1675, he found the equation ycbc =i dxy -^ xdy, 
giving an expression for d{xy)y which he observed to be true 
for all curves. He succeeded also in eliminating dx from 
a differential equation, so that it contained only dy, and 
thereby led to the solution of the problem under considera- 
tion. " Behold, a most elegant way by which the problems 
of the inverse methods of tangents are solved, or at least 
are reduced to quadratures ! " Thus he saw clearly that the 
inverse problems of tangents could be solved by quadratures, 
or, in other words, by the integral calculus. In course of a 
half-year he discovered that the direct problem of tangents, 
too, yielded to the power of his new calculus, and that thereby 
a more general solution than that of Descartes could be 
obtained. He succeeded in solving all the special problems 
of this kind, which had been left unsolved by Descartes. 
Of these we mention only the celebrated problem proposed 
to Descartes by De Beaune, viz. to find the curve whose 
ordinate is to its sub-tangent as a given line is to that part 
of the ordinate which lies between the curve and a line drawn 
from the vertex of the curve at a given inclination to the axis. 

Such was, in brief, the progress in the evolution of the new 
calculus made by Leibniz during his stay in Paris. Before 
his departure, in October, 1676, he found himself in possession 
of the most elementary rules and formula of the infinitesimal 
calculus. 

From Paris, Leibniz returned to Hanover by way of London 
and Amsterdam. In London he met Collins, who showed him 



224 A HISTORY OF MATHEMATICS. 

a part of his scientific correspondence. Of this we shall speak 
later. In Amsterdam he discussed mathematics with Sluze, 
and became satisfied that his own method of constructing 
tangents not only accomplished all that Sluze's did, but even 
more, since it could be extended to three variables, by which 
tangent planes to surfaces could be found; and especially, 
since neither irrationals nor fractions prevented the immediate 
application of his method. 

In a paper of July 11, 1677, Leibniz gave correct rules for 
the differentiation of sums, products, quotients, powers, and 
roots. He had given the differentials of a few negative and 
fractional powers, as early as November, 1676, but had made 
some mistakes. For d -y/x he had given the erroneous value 

— —, and in another place the value — ^a?"^; for d-^ occurs in 

Vx 2 

one place the wrong value, — —, while a few lines lower is 

given — --, its correct value. 

In 1682 was founded in Berlin the Acta Eruditorum, a 
journal usually known by the name of Leipzig Acts, It 
was a partial imitation of the French Journal dee Savans 
(founded in 1665), and the literary and scientific review 
published in Germany. Leibniz was a frequent contributor. 
Tschirnhaus, who had studied mathematics in Paris with 
Leibniz, and who was familiar with the new analysis of 
Leibniz, published in the Acta Eroditorum a paper on quad- 
ratures, which consists principally of subject-matter com- 
municated by Leibniz to Tschirnhaus during a controversy 
which they had had on this subject. Fearing that Tschirnhaus 
might claim as his own and publish the notation and rules of 
the differential calculus, Leibniz decided, at last, to make 
public the fruits of his inventions. In 1684, or nine years 
after the new calculus first dawned upon the mind of Leibniz, 
and nineteen years after Newton first worked at fiuxions, 



NEWTON TO EULBB. 226 

and three years before the publication of Newton's Principui, 
Leibniz published^ in the Leipzig Acts, his first paper on the 
differential calculus. He was unwilling to give to the world 
all his treasures, but chose those parts of his work which were 
most abstruse and least perspicuous. This epoch-making paper 
of only six pages bears the title : '^ Kova methodus pro maximis 
et minimis, itemque tangentibus, quae nee fractas nee irra- 
tionales quantitates moratur, et singulare pro illis calculi 
genus." The rules of calculation are briefly stated without 
proof, and the meaning of dz and dy is not made clear. It has 
been inferred from this that Leibniz himself had no definite 
and settled ideas on this subject. Are dy and dx finite or 
infinitesimal quantities ? At first they appear, indeed, to have 
been taken as finite, when he says : '^ We now call any line 
selected at random dx, then we designate the line which is to 
da; as 2/ is to the sub-tangent, by dy, which is the difference of 
y" Leibniz then ascertains, by his calculus, in what way a 
ray of light passing through two differently refracting media, 
can travel easiest from one point to another ; and then closes 
his article by giving his solution, in a few words, of De 
Beaune's problem. Two years later (1686) Leibniz published 
in the Acta Eruditorum a paper containing the rudiments of 
the integral calculus. The quantities dx and dy are there 
treated as infinitely small. He showed that by the use of his 
notation, the properties of curves could be fully expressed by 
equations. Thus the equation 

, r dx 

characterises the cycloid." 

The great invention of Leibniz, now made public by his 
articles in the Leipzig Acta, made little impression upon the 
mass of mathematicians. In Qermany no one comprehended 



226 A HISTORY OF MATHEMATICS. 

the new calculus except Tschirnhaus, who remained indif- 
ferent to it. The author's statements were too short and suc- 
cinct to make the calculus generally understood. The first 
to recognise its importance and to take up the study of it 
were two foreigners, — the Scotchman ITiomas Craige, and 
the Swiss James Bernoulli. The latter wrote Leibniz a 
letter in 1687, wishing to be initiated into the mysteries 
of the new analysis. Leibniz was then travelling abroad, 
so that this letter remained unanswered till 1790. James 
Bernoulli succeeded, meanwhile, by close application, in un- 
covering the secrets of the differential calculus without assist- 
ance. He and his brother John proved to be mathematicians 
of exceptional power. They applied themselves to the new 
science with a success and to an extent which made Leibniz 
declare that it was as much theirs as his. Leibniz carried on 
an extensive correspondence with them, as well as with other 
mathematicians. In a letter to John Bernoulli he suggests, 
among other things, that the integral calculus be improved by 
reducing integrals back to certain fundamental irreducible 
forms. The integration of logarithmic expressions was then 
studied. The writings of Leibniz contain many innovations, 
and anticipations of since prominent methods. Thus he made 
use of variable parameters, laid the foundation of analysis in 
situ, introduced the first notion of determinants in his effort 
to simplify the expression arising in the elimination of the 
unknown quantities from a set of linear equations. He 
resorted to the device of breaking up certain fractions into 
the sum of other fractions for the purpose of easier integration ; 
he explicitly assumed the principle of continuity ; he gave the 
first instance of a " singular solution/' and laid the foundation 
to the theory of envelopes in two papers, one of which contains 
for the first time the terms co-ordinate and axes of co-ordinates. 
He wrote on osculating curves, but his paper contained the 



NEWTON TO EULEE. 227 

error (pointed out by John Bernoulli^ but not admitted by 
him) that an osculating circle will necessarily cut a curve in 
four consecutive points. Well known is his theorem on the 
nth differential coefficient of the product of two functions of a 
variable. Of his many papers on mechanics, some are valuable, 
while others contain grave errors. 

Before tracing the further development of the calculus we 
shall sketch the history of that long and bitter controversy 
between English and Continental mathematicians on the inven- 
tion of the calculus. The question was, did Leibniz invent it 
independently of Newton, or was he a plagiarist ? 

We must begin with the early correspondence between the 
parties appearing in this dispute. Newton had begun using 
his notation of fluxions in 1666.^ In 1669 Barrow sent Collins 
Newton's tract, De AncUyai per Equationesy etc. 

The first visit of Leibniz to London extended from the 11th 
of January until March, 1673. He was in the habit of com- 
mitting to writing important scientific communications received 
from others. In 1890 Grerhardt discovered in the royal library 
at Hanover a sheet of manuscript with notes taken by Leibniz 
during this journey.** They are headed " Observata Philoso- 
phica in itinere Anglicano sub initium anni 1673." The sheet 
is divided by horizontal lines into sections. The sections 
g^ven to Chymica, Mechanica, Magnetica, Botanica, Anatomica, 
Medica, Miscellanea, contain extensive memoranda, while those 
devoted to mathematics have very few notes. Under Geo- 
metrica he says only this : " Tangentes omnium figurarum. 
Figurarum geometricarum explicatio per motum puncti in 
moto lati." We suspect from this that Leibniz had read 
Barrow's lectures. Newton is referred to only under Optica. 
^Evidently Leibniz did not obtain a knowledge of fluxions 
during this visit to London, nor is it claimed that he did by 
his opponents. 



228 A HISTORY OF MATHEMATICS. 

Various letters of Kewton^ Collins^ and others, up to the 
beginning of 1676, state that Newton invented a method by 
which tangents could be drawn without the necessity of freeing 
their equations from irrational terms. Leibniz announced in 
1674 to Oldenburg, then secretary of the Royal Society, that 
he possessed very general analytical methods, by which he had 
found theorems of great importance on the quadrature of the 
circle by means of series. In answer, Oldenburg stated 
Newton and James Gregory had also discovered methods of 
quadratures, which extended to the circle. Leibniz desired to 
have these methods communicated to him; and Newton, at 
the request of Oldenburg and Collins, wrote to the former the 
celebrated letters of June 13 and October 24, 1676. The first 
contained the Binomial Theorem and a variety of other mat- 
ters relating to infinite series and quadratures; but nothing 
directly on the method of fluxions. Leibniz in reply speaks 
in the highest terms of what Newton had done, and requests 
further explanation. Newton in his second letter just men- 
tioned explains the way in which he found the Binomial 
Theorem, and also communicates his method of fluxions and 
fluents in form of an anagram in which all the letters in the 
sentence communicated were placed in alphabetical order. 
Thus Newton says that his method of drawing tangents was 

6accdoe ISeff li 31 9n 4o 4grr 4s 9t 12'ua?. 

The sentence was, "Data sequatione quotcunque fluentes 
quantitates involvente fluxiones in venire, et vice versa." 
("Having any given equation involving never so many flowing 
quantities, to find the fluxions, and vice versa.") Surely this 
anagram afforded no hint. Leibniz wrote a reply to Collins, 
in which, without any desire of concealment, he explained the 
principle, notation, and the use of the differential calculus. 
The death of Oldenburg brought this correspondence to a 



KEWTOK TO EULEB. 229 

dose. Nothing material happened till 1684, when Leibniz 
published his first paper on the differential calculus in the 
Leipzig Acts, so that while Newton's claim to the priority of 
inyention must be admitted by all, it must also be granted 
that Leibniz was the first to give the full benefit of the calcu- 
lus to the world. Thus, while Newton's invention remained a 
secret, communicated only to a few friends, the calculus of 
Leibniz was spreading over the Continent. No rivalry or 
hostility existed, as yet, between the illustrious scientists. 
Newton expressed a very favourable opinion of Leibniz's 
inventions, known to him through the above correspondence 
with Oldenburg, in the following celebrated scholium {Prind- 
piOj first edition, 1687, Book II., Prop. 7, scholium): — 

" In letters which went between me and that most excellent 
geometer, Q. G. Leibniz, ten years ago, when I signified that 
I was in the knowledge of a method of determining maxima 
and minima, of drawing tangents, and the like, and when I 
concealed it in transposed letters involving this sentence (Data 
sequatione, etc., above cited), that most distinguished man 
wrote back that he had also fallen upon a method of the same 
kind, and communicated his method, which hardly differed 
from mine, except in his forms of words and symbols." 

As regards this passage, we shall see that Newton was after- 
wards weak enough, as De Morgan says : " First, to deny the 
plain and obvious meaning, and secondly, to omit it entirely 
from the third edition of the Principia,^^ On the Continent, 
great progress was made in the calculus by Leibniz and his 
coadjutors, the brothers James and John Bernoulli, and 
Marquis de I'Hospital. In 1695 Wallis informed Newton by 
letter that '< he had heard that his notions of fluxions passed 
in Holland with great applause by the name of 'Leibniz's 
Calculus Differentialis.'" Accordingly Wallis stated in the 
preface to a volume of his works that the calculus differen- 



230 A HISTORY OF MATHEMATICS. 

tialis was Newton's method of fluxions which had been 
communicated to Leibniz in the Oldenburg letters. A review 
of Wallis' works^ in the Leipzig Acts for 1696, reminded the 
reader of Newton's own admission in the scholium above 
cited. 

For fifteen years Leibniz had enjoyed unchallenged the 
honour of being the inventor of his calculus. But in 1699 Fato 
de Duillier, a Swiss, who had settled in England, stated in a 
mathematical paper, presented to the Koyal Society, his con- 
viction that Newton was the first inventor; adding that, 
whether Leibniz, the second inventor, had borrowed anything 
from the other, he would leave to the judgment of those who 
had seen the letters and manuscripts of Newton. This was 
the first distinct insinuation of plagiarism. It would seem that 
the English mathematicians had for some time been cherishing 
suspicions unfavourable to Leibniz. A feeling had doubtless 
long prevailed that Leibniz, during his second visit to London 
in 1676, had or might have seen among the papers of Collins, 
Newton's Analysis per cequationeSy etc., which contained appli- 
cations of the fluxionary method, but no systematic develop- 
ment or explanation of it. Leibniz certainly did see at least 
part of this tract. During the week spent in London, he took 
note of whatever interested him among the letters and papers 
of Collins. His memoranda discovered by Gerhardt in 1849 in 
the Hanover library fill two sheets. ^ The one bearing on our 
question is headed "Excerpta ex tractatu Newtoni Msc. de 
Analysi per aequationes numero terminorum infinitas." The 
notes are very brief, excepting those De Resolutione ceqtia- 
tionum affectarurriy of which there is an almost complete copy. 
This part was evidently new to him. If he examined 
Newton's entire tract, the other parts did not particularly 
impress him. From it he seems to have gained nothing per- 
taining to the infinitesimal calculus. By the previous intro- 



NEWTON TO EULBB. 281 

dnction of his own algorithm he had made greater progress 
than by what came to his knowledge in London. Nothing 
mathematical that he had received engaged his thoughts in 
the immediate future^ for on his way back to Holland he com- 
posed a lengthy dialogue on mechanical subjects. 

Doillier's insinuations lighted up a flame of discord which a 
whole century was hardly sufficient to extinguish. Leibniz, 
who had never contested the priority of Newton's discovery, 
and who appeared to be quite satisfied with Newton's admis- 
sion in his scholium, now appears for the first time in the 
controversy. He made an animated reply in the Leipzig Acts, 
and complained to the Koyal Society of the injustice done him. 

Here the affair rested for some time. In the Q^adrcUure of 
Curves, published 1704, for the first time^ a formal exposition 
of the method and notation of fluxions was made public. In 
1705 appeared an unfavourable review of this in the Leipzig 
Acts, stating that Newton uses and always has used fluxions 
for the differences of Leibniz. This was considered by New- 
ton's friends an imputation of plagiarism on the part of their 
chief, but this interpretation was always strenuously resisted 
by Leibniz. Keill, professor of astronomy at Oxford, under- 
took with more zeal than judgment the defence of Newton. 
In a paper inserted in the Philosophical Transactions of 1708, 
he claimed that Newton was the first inventor of fluxions and 
'' that the same calculus was afterward published by Leibniz, 
the name and the mode of notation being changed." Leibniz 
complained to the secretary of the Royal Society of bad treat- 
ment and requested the interference of that body to induce 
Keill to disavow the intention of imputing fraud. Keill was 
not made to retract his accusation; on the contrary, was 
authorised by Newton and the Royal Society to explain and 
defend his statement. This he did in a long letter. Leibniz 
thereupon complained that the charge was now more open than 



232 A HISTORY OF liATHBMATICS. 

before, and appealed for justice to the Boyal Society and to 
Newton himself. The Boyal Society, thus appealed to as a 
judge, appointed a committee which collected and reported 
upon a large mass of documents — mostly letters from and to 
Newton, Leibniz, Wallis, Collins, etc. This re|)ort, called the 
Commercium Epistolicum, appeared in the year 1712 and again 
in 1725, with a Kecensio prefixed, and additional notes by KeiU. 
The final conclusion in the Commercium Epistolicum was 
that Newton was the first inventor. But this was not to the 
point. The question was not whether Newton was the first 
inventor, but whether Leibniz had stolen the method. The 
committee had not formally ventured to assert their belief 
that Leibniz was a plagiarist. Yet there runs throughout the 
document a desire of proving Leibniz guilty of more than 
they meant positively to affirm. Leibniz protested only in 
private letters against the proceeding of the Koyal Society, 
declaring that he woiild not answer an argument so weak. 
John Bemoiilli, in a letter to Leibniz, which was published 
later in an anonymous tract, is as decidedly unfair towards 
Newton as the friends of the latter had been towards Leibniz. 
Keill replied, and then Newton and Leibniz appear as mutual 
accusers in several letters addressed to third parties. In a 
letter to Conti, April 9, 1716, Leibniz again reminded Newton 
of the admission he had made in the scholium, which he was 
now desirous of disavowing; Leibniz also states that he 
always believed Newton, but that, seeing him connive at 
accusations which he must have known to be false, it was 
natural that he (Leibniz) should begin to doubt. Newton 
did not reply to this letter, but circulated some remarks among 
his friends which he published immediately after hearing 
of the death of Leibniz, November 14, 1716. This paper 
of Newton gives the following explanation pertaining to the 
scholium in question: '^He [Leibniz] pretends that in my 



NEWTON TO EULBR. 283 

book of principles I allowed him the invention of the eaJculns 
differentialis^ independently of my own ; and that to attribute 
this invention to myself is contrary to my knowledge there 
avowed. But in the paragraph there referred unto I do not 
find one word to this purpose/' In the third edition of the 
Prindpia, 1725, Newton omitted the scholium and substituted 
in its place another, in which the name of Leibniz does not 
appear. 

National pride and party feeling long prevented the adoption 
of impartial opinions in England, but now it is generally ad- 
mitted by nearly all familiar with the matter, that Leibniz 
really was an independent inventor. Perhaps the most tell- 
ing evidence to show that Leibniz was an independent inven- 
tor is found in the study of his mathematical papers (collected 
and edited by C. I. Gerhardt, in six volumes, Berlin, 1849- 
1860), which point out a gradual and natural evolution of the 
rules of the calculus in his own mind. '< There was through- 
out the whole dispute," says De Morgan, ^' a confusion between 
the knowledge of fluxions or differentials and that of a ccUcti- 
lu8 of fluxions or differentials ; that is, a digested method with 
general rules." 

This controversy is to be regretted on account of the long 
and bitter alienation which it produced between English and 
Continental mathematicians. It stopped almost completely 
all interchange of ideas on scientific subjects. The English 
adhered closely to Newton's methods and, until about 1820, 
remained, in most cases, ignorant of the brilliant mathematical 
discoveries that were being made on the Continent. The loss 
in point of scientific advantage was almost entirely on the 
side of Britain. The only way in which this dispute may be 
said, in a small measure, to have furthered the progress of 
mathematics, is through the challenge problems by which 
each side attempted to annoy its adversaries. 



234 A HISTORY OF MATHEMATICS. 

The recurring practice of issuing challenge problems was 
inaugurated at this time by Leibniz. They were, at first, not 
intended as defiances, but merely as exercises in the new cal- 
culus. Such was the problem of the isochronous curre (to 
find the curve along which a body falls with uniform velocity), 
proposed by him to the Cartesians in 1687, and solved by 
James Bernoulli, himself, and John Bernoulli. James Ber- 
noulli proposed in the Leipzig Journal the question to find the 
curve (the catenary) formed by a chain of uniform weight 
suspended freely from its ends. It was resolved by Huygens, 
Leibniz, and himself. In 1697 John Bernoulli challenged the 
best mathematicians in Europe to solve the difficult problem, 
to find the curve (the cycloid) along which a body falls from 
one point to another in the shortest possible time. Leibniz 
solved it the day he received it. Newton, de I'Hospital, and 
the two Bemoullis gave solutions. Newton's appeared anony- 
mously in the PhUosqphiccU Transactions, but John Bernoulli 
recognised in it his powerful mind, " tanquam," he says, " ex 
ungue leonem." The problem of orthogonal trajectories (a 
system of curves described by a known law being given, to 
describe a curve which shall cut them all at right angles) had 
been long proposed in the Acta Eruditorum, but failed at 
first to receive much attention. It was again proposed in 
1716 by Leibniz, to feel the pulse of the English mathema- 
ticians. 

This may be considered as the first defiance problem pro- 
fessedly aimed at the English. Newton solved it the same 
evening on which it was delivered to him, although he was 
much fatigued by the day's work at the mint. His solution, 
as published, was a general plan of an investigation rather 
than an actual solution, and was, on that account, criticised by 
Bernoulli as being of no value. Brook Taylor undertook the 
defence of it, but ended by using very reprehensible language. 



NEWTON TO BULBB. 286 

Bernoulli was not to be outdone in incivility, and made a 
bitter reply. Not long afterwards Taylor sent an open de- 
fiance to Continental mathematicians of a problem on the 
integration of a fluxion of complicated form which was known 
to very few geometers in England and supposed to be beyond 
the power of their adversaries. The selection was injudicious, 
for Bernoulli had long before explained the method of this 
and similar integrations. It served only to display the skill 
and augment the triumph of the followers of Leibniz. The 
last and most unskilful challenge was by John KeilL The 
problem •was to find the path of a projectile in a medium 
which resists proportionally to the square of the velocity. 
Without first making sure that he himself could solve it, 
Keill boldly challenged Bernoulli to produce a solution. The 
latter resolved the question in very short time, not only for a 
resistance proportional to the square, but to any power of the 
velocity. Suspecting the weakness of the adversary, he re- 
peatedly offered to send his solution to a confidential person 
in London, provided Keill would do the same. Keill never 
made a reply, and Bernoulli abused him and cruelly exulted 
over him.* 

The explanations of the fundamental principles of the cal- 
culus, as given by Newton and Leibniz, lacked clearness and 
rigour. For that reason it met with opposition from several 
quarters. In 1694 Bernard Nieuwentyt of Holland denied 
the existence of differentials of higher orders and objected to 
the practice of neglecting infinitely small quantities. These 
objections Leibniz was not able to meet satisfactorily. In his 

reply he said the value of -^ in geometry could be expressed 

cue 

as the ratio of finite quantities. In the interpretation of dx 

and dy Leibniz vacillated. At one time they appear in his 

writings as finite lines ; then they are called infinitely small 



236 A HISTORY OF MATHEMATICS. 

quantities, and again, qttantUates incbsaignabiles, which spring 
from quantitates assigndbiles by the law of continuity. In this 
last presentation Leibniz approached nearest to Newton. 

In England the principles of fluxions were boldly attacked 
by Bishop Berkeley, the eminent metaphysician, who argued 
with great acuteness, contending, among other things, that 
the fundamental idea of supposing a finite ratio to exist 
between terms absolutely evanescent — "the ghosts of de- 
parted quantities," as he called them — was absurd and unin- 
telligible. The reply made by Jurin failed to remove all the 
objections. Berkeley was the first to point out what was 
again shown later by Lazare Carnot, that correct answers were 
reached by a "compensation of errors." Berkeley's attack 
was not devoid of good results, for it was the immediate cause 
of the work on fluxions by Maclaurin. In France Michel 
Rolle rejected the differential calculus and had a controversy 
with Vdrignon on the subject. 

Among the most vigorous promoters of the calculus on the 
Continent were the Bernoullis. They and Euler made Basel 
in Switzerland famous as the cradle of great mathematicians. 
The family of Bernoullis furnished in course of a century 
eight members who distinguished themselves in mathematics. 
We subjoin the following genealogical table : — 

Nicolaus Bernoulli^ the Father 



Jacob, 1664-1706 Nicolaus Johann, 1667-1748 

Nicolau8, 1687-1769 Nicolaus, 1695-1726 

Daniel, 1700-1782 
Johann, 1710-1790 



Daniel Johann, 1744-1807 Jacob, 1768-1789 

Most celebrated were the two brothers Jacob (James) and 
Johann (John), and Daniel, the son of John. James and 



NEWTON TO EULKB. 287 

John were staunch friends of Leibniz and worked hand in 
hand with him. James Bernoulli (1654-1705) was bom in 
Basel. Becoming interested in the calculus, he mastered it 
without aid from a teacher. From 1687 until his death he 
occupied the mathematical chair at the University of BaseL 
He was the first to give a solution to Leibniz's problem of the 
isochronous curve. In his solution, published in the Acta 
Erudilorum, 1690, we meet for the first time with the word 
integral Leibniz had called the integral calculus cakvlus 
summatoriuSf but in 1696 the term calculus integralis was 
agreed upon between Leibniz and John Bernoulli. James 
proposed the problem of the catenary, then proved the correct- 
ness of Leibniz's construction of this curve, and solved the 
more complicated problems, supposing the string to be (1) of 
variable density, (2) extensible, (3) acted upon at each point 
by a force directed to a fixed centre. Of these problems he 
published answers without explanations, while his brother 
John gave in addition their theory. He determined the shape 
of the " elastic curve " formed by an elastic plate or rod fixed 
at one end and bent by a weight applied to the other end ; of 
the "lintearia," a flexible rectangular plate with two sides 
fixed horizontally at the same height, filled with a liquid ; of 
the " volaria," a rectangular sail filled with wind. He studied 
the loxodromic and logarithmic spirals, in the last of which 
he took particular delight from its remarkable property of 
reproducing itself under a variety of conditions. Following 
the example of Archimedes, he willed that the curve be en- 
graved upon his tombstone with the inscription " eadem mutata 
reaurgo" In 1696 he proposed the famous problem of isoper- 
imetrical figures, and in 1701 published his own solution. He 
wrote a work on Ars Conjectandi, which is a development of 
the calculus of probabilities and contains the investigation 
now called ^^Bernoulli's theorem" and the so-called '< numbers 



238 A HISTORY OF MATHEMATICS. 

of Bernoulli/' which are in fact (though not so considered by 

him) the coefficients of — in the expansion of (e*— 1)~*. Of 

n! 

his collected works^ in three volumes, one was printed in 1713, 

the other two in 1744. 

John Bemoolli (1667-1748) was initiated into mathematics 
by his brother. He afterwards visited France, where he met 
Malebranche, Cassini, De Lahire, Varignon, and de I'Hospital. 
For ten years he occupied the mathematical chair at Groningen 
and then succeeded his brother at Basel. He was one of the 
most enthusiastic teachers and most successful original inves- 
tigators of his time. He was a member of almost every learned 
society in Europe. His controversies were almost as numerous 
as his discoveries. He was ardent in his friendships, but 
tmfair, mean, and violent toward all who incurred his dislike 
•—even his own brother and son. He had a bitter dispute 
with James on the isoperimetrical problem. James convicted 
him of several paralogisms. After his brother's death he 
attempted to substitute a disguised solution of the former for 
an incorrect one of his own. John admired the merits of 
Leibniz and Euler, but was blind to those of Newton. He 
immensely enriched the integral calculus by his labours. 
Among his discoveries are the exponential calculus, the line 
of swiftest descent, and its beautiful relation to the path 
described by a ray passing through strata of variable density. 
He treated trigonometry by the analytical method, studied 
caustic curves and trajectories. Several times he was given 
prizes by the Academy of Science in Paris. 

Of his sons, Nicholas and Daniel were appointed professors 
of mathematics at the same time in the Academy of St. 
Petersburg. The former soon died in the prime of life; the 
latter returned to Basel in 1733, where he assumed the chair 
of experimental philosophy. His first mathematical publi- 



NBWTON TO BULSB. 289 

cation was the solution of a differential equation proposed by 
Biccati. He wrote a work on hydrodynamics. His investigdr 
tions on probability are remarkable for their boldness and 
originality. He proposed the theory of morcU expectation^ 
which he thought would give results more in accordance with 
our ordinary notions than the theory of matJiematiccd prob- 
ability. His '^ moral expectation '' has become classic, but no 
one ever makes use of it. He applies the theory of probability 
to insurance ; to determine the mortality caused by small-pox 
at various stages of life ; to determine the number of survivors 
at a given age from a given number of births ; to determine 
how much inoculation lengthens the average duration of life. 
He showed how the differential calculus could be used in the 
theory of probability. He and Euler enjoyed the honour of 
having gained or shared no less than ten prizes from the 
Academy of Sciences in Paris. 

Johann Bernoulli (bom 1710) succeeded his father in the 
professorship of mathematics at Basel. He captured three 
prizes (on the capstan, the propagation of light, and the 
magnet) from the Academy of Sciences at Paris. Nlcolaos 
Bemoolli (born 1687) held for a time the mathematical chair 
at Padua which Galileo had once filled. Johann Bernoulli 
(bom 1744) at the age of nineteen was appointed astronomer 
royal at Berlin, and afterwards director of the mathematical 
department of the Academy. His brother Jacob took upon 
himself the duties of the chair of experimental physics at 
Basel, previously performed by his uncle Jacob, and later 
was appointed mathematical professor in the Academy at St 
Petersburg. 

Brief mention will now be made of some other mathemati- 
cians belonging to the period of Newton, Leibniz, and the 
elder Bernoullis. 

Quillaume Franqoia Antoine THoipital (1661-1704), a pupil 



240 A HISTOBY OF MATHEMATICS. 

of John Bernoulli, has already been mentioned as taking 
part in the challenges issued by Leibniz and the Bemoullis. 
He helped powerfully in making the calculus of Leibniz better 
known to the mass of mathematicians by the publication of a 
treatise thereon in 1696. This contains for the first time the 
method of finding the limiting value of a fraction whose two 
terms tend toward zero at the same time. 

Another zealous French advocate of the calculus was Pierre 
Varignon (1654-1722). Joseph Saurin (1669-1737) solved the 
delicate problem of how to determine the tangents at the 
multiple points of algebraic curves. Franqois Nicole (1683- 
1758) in 1717 issued the first systematic treatise on finite 
differences, in which he finds the sums of a considerable 
number of interesting series. He wrote also on roulettes, 
particularly spherical epicycloids, and their rectification. Also 
interested in finite differences was Pierre Raymond de Hontmort 
(1678-1719). His chief writings, on the theory of probabil- 
ity, served to stimulate his more distinguished successor, De 
Moivre. Jean Paul de Gua (1713-1785) gave the demonstration 
of Descartes' rule of signs, now given in books. This skilful 
geometer wrote in 1740 a work on analytical geometry, the 
object of which, was to show that most investigations on curves 
could be carried on with the analysis of Descartes quite as 
easily as with the calculus. He shows how to find the tan- 
gents, asymptotes, and various singular points of curves of all 
degrees, and proved by perspective that several of these points 
can be at infinity. A mathematician who clung to the methods 
of the ancients was Philippe de Lahire (1640-1718), a pupil of 
Desargues. His work on conic sections is purely synthetic, 
but differs from ancient treatises in deducing the properties of 
conies from those of the circle in the same manner as did 
Desargues and Pascal. His innovations stand in close relation 
with modern synthetic geometry. He wrote on roulettes, on 



NEWTON TO EULER. 241 

graphical methods, epicycloids, conchoids, and on magic 
squares. Ifichel Rolle (1652-1719) is the author of a theorem 
named after him. 

Of Italian mathematicians, Riccati and Fagnano must not 
remain unmentioned. Jacopo Francesco, Count Riccati (1676- 
1754) is best known in connection with his problem, called 
Riccati's equation, published in the Acta Eruditorum in 1724. 
He succeeded in integrating this differential equation for some 
special cases. A geometrician of remarkable power was Giulio 
Carlo, Count de Fagnano (1682-1766). He discovered the fol- 

lowing formula, 7r=2ilog--^^j in which he anticipated Euler 

1-f I 

in the use of imaginary exponents and logarithms. His studies 
on the rectification of the ellipse and hyperbola are the start- 
ing-points of the theory of elliptic functions. He showed, for 
instance, that two arcs of an ellipse can be found in an in- 
definite number of ways, whose difference is expressible by a 
right line. 

In Germany the only noted contemporary of Leibniz is 
Ehrenfried Walter Tschimhausen (1631-1708), who discovered 
the caustic of reflection, experimented on metallic reflectors 
and large burning-glasses, and gave us a method of transform- 
ing equations named after him. Believing that the most 
simple methods (like those of the ancients) are the most 
correct, he concluded that in the researches relating to the 
properties of curves the calculus might as well be dispensed 
with. 

After the death of Leibniz there was in (Jermany not a 
single mathematician of note. Christian Wolf (1679-1754), 
professor at Halle, was ambitious to figure as successor of 
Leibniz, but he " forced the ingenious ideas of Leibniz into a 
pedantic scholasticism, and had the unenviable reputation of 
having presented the elements of the arithmetic, algebra, and 



242 A HISTORY OF MATHEMATICS. 

analysis developed since the time of the Renaissance in the 
form of Euclid, — of course only in outward form, for into the 
spirit of them he was quite unable to penetrate." *• 

The contemporaries and immediate successors of Newton in 
Great Britain were men of no mean merit. We have refer- 
ence to Cotes, Taylor, Maclaurin, and De Moivre. We are 
told that at the death' of Roger Cotes (1682-1716), Newton 
exclaimed, '^ If Cotes had lived, we might have known some- 
thing." It was at the request of Dr. Bentley that Cotes 
undertook the publication of the second edition of Newton's 
Principia, His mathematical papers were published after his 
death by Robert Smith, his successor in the Plumbian pro- 
fessorship at Trinity College. The title of the work, Har- 
monia Mensurarumf was suggested by the following theorem 
contained in it : If on each radius vector, through a fixed point 
O, there be taken a point R, such that the reciprocal of OR be 
the arithmetic mean of the reciprocals of ORi, OR^ ••• OR^ 
then the locus of R will be a straight line. In this work 
progress was made in the application of logarithms and the 
properties of the circle to the calculus of fluents. To Cotes 
we owe a theorem in trigonometry which depends on the 
forming of factors of ic" — 1. Chief among the admirers of 
Newton were Taylor and Maclaurin. The quarrel between 
English and Continental mathematicians caused them to work 
quite independently of their great contemporaries across the 
Channel. 

Brook Taylor (1685-1731) was interested in many branches 
of learning, and in the latter part of his life engaged mainly in 
religious and philosophic speculations. His principal work, 
Methodus incrementorum directa et inversa, London, 1715-1717, 
added a new branch to mathematics, now called " finite differ- 
ences." He made many important applications of it, par- 
ticularly to the study of the form of movement of vibrating 



NEWTON TO EULBB. 2i8 

strings, first reduced to mechanical principles by him. This 
work contains also "Taylor's theorem," the importance of 
which was not recognised by analysts for over fifty years, 
until Lagrange pointed out its power. His proof of it does not 
consider the question of convergency, and is quite worthless. 
The first rigorous proof was given a century later by Cauchy. 
Taylor's work contains the first correct explanation of astro- 
nomical refraction. He wrote also a work on linear per- 
spective, a treatise which, like his other writings, suffers for 
want of fulness and clearness of expression. At the age of 
twenty-three he gave a remarkable solution of the problem of 
the centre of oscillation, published in 1714. His claim to 
priority was unjustly disputed by John Bernoulli. 

Colin Bfadaurin (1698-1746) was elected professor of mathe- 
matics at Aberdeen at the age of nineteen by competitive 
examination, and in 1725 succeeded James Gregory at the Uni- 
versity of Edinburgh. He enjoyed the friendship of Newton, 
and, inspired by Newton's discoveries, he published in 1719 his 
Oeomelria Organicay containing a new and remarkable mode 
of generating conies, known by his name. A second tracts 
De Linearum yeometricarum Proprietatibus, 1720, is remarkable 
for the elegance of its demonstrations. It is based upon two 
theorems: the first is the theorem of Cotes; the second is 
Maclaurin's : If through any point O a line be drawn meeting 
the curve in n points, and at these points tangents be drawn, 
and if any other line through O cut the curve in i?„ Rf, etc., 

and the system of n tangents in r^, r,, etc., then 2775 = 277"- 

%This and Cotes' theorem are generalisations of theorems of 
Newton. Maclaurin uses these in his treatment of curves of 
the second and third degree, culminating in the remarkable 
theorem that if a quadrangle has its vertices and the two 
points of intersection of its opposite sides upon a curve of the 



/ 



244 A HISTORY OF MATHEMATICS. 

third degree, then the tangents drawn at two opposite yertices 
cut each other on the curve. He deduced independently 
Pascal's theorem on the hexagram. The following is his ex- 
tension of this theorem {Phil. Trans., 1735) : If a polygon 
move so that each of its sides passes through a fixed point, 
and if all its summits except one describe curves of the degrees 
m, n, p, etc., respectively, then the free summit moves on a 
curve of the degree 2 mnp"*, which reduces to mnp-" when 
the fixed points all lie on a straight line. Maclaurin wrote on 
/\ pedal curves. He is the author of an Algebra. The object of 
his treatise on Fluxions was to found the doctrine of fluxions 
on geometric demonstrations after the manner of the ancients, 
and thus, by rigorous exposition, answer such attacks as Berke- 
ley's that the doctrine rested on false reasoning. The Fluxions 
contained for the first time the correct way of distinguishing 
between maxima and minima, and explained their use in the 
theory of multiple points. "Maclaurin's theorem" was pre- 
viously given by James Stirling, and is but a particular case 
of " Taylor's theorem." Appended to the treatise on Fluxions 
is the solution of a number of beautiful geometric, mechanical, 
and astronomical problems, in which he employs ancient 
methods with such consummate skill as to induce Clairaut to 
abandon analytic methods and to attack the problem of the 
figure of the earth by pure geometry. His solutions com- 
manded the liveliest admiration of Lagrange. Maclaurin in- 
vestigated the attraction of the ellipsoid of revolution, and 
showed that a homogeneous liquid mass revolving uniformly 
around an axis under the action of gravity must assume the 
form of an ellipsoid of revolution. Newton had given this • 
theorem without proof. Notwithstanding the genius of Mac- 
laurin, his influence on the progress of mathematics in Great 
Britain was unfortunate; for, by his example, he induced his 
countrymen to neglect analysis and to be indifferent to the 



NEWTON TO EULBB. 245 

wonderful progress in the higher analysis made on the Con- 
tinent. 

It remains for us to speak of Abraham de Moivre (1667-1754), 
who was of French descent, but was compelled to leave France 
at the age of eighteen, on the Eevocation of the Edict of Nantes. 
He settled in London, where he gave lessons in mathematics. 
He lived to the advanced age of eighty-seven and sank into a 
state of almost total lethargy. His subsistence was latterly 
dependent on the solution of questions on games of chance and 
problems on probabilities, which he was in the liabit of giving 
at a tavern in St. Martin's Lane. Shortly before his death he 
declared that it was necessary for him to sleep ten or twenty 
minutes longer every day. The day after he had reached the 
total of over twenty-three hours, he slept exactly twenty-four 
hours and then passed away in his sleep. De Moivre enjoyed 
the friendship of Newton and Halley. His power as a math- 
ematician lay in analytic rather than geometric investigation. 
He revolutionised higher trigonometry by the discovery of the 
theorem known by his name and by extending the theorems on 
the multiplication and division of sectors from the circle to the 
hyperbola. His work on the theory of probability surpasses 
anything done by any other mathematician except Laplace. 
His principal contributions are his investigations respecting 
the Duration of Play, his Theory of Recurring Series, and 
his extension of the value of Bernoulli's theorem by the aid 
of Stirling's theorem.** His chief works are the Doctrine of 
Chances, 1716, the Miscellanea Analytica, 1730, and his papers 
in the Philosophical Transactions. 



246 A HISTORY OF MATHEMATICS. 



EULER, LAGRANGE, AND LAPLACE. 

During the epoch of ninety years from 1730 to 1820 the French 
and Swiss cultivated mathematics with most brilliant success. 
No previous period had shown such an array of illustrious 
names. At this time Switzerland had her Euler ; France, her 
Lagrange, Laplace, Legendre, and Monge. The mediocrity of 
French mathematics which marked the time of Louis XIY. 
was now followed by one of the very brightest periods of all 
history. England and Germany, on the other hand, which 
during the unproductive period in France had their Newton 
and Leibniz, could now boast of no great mathematician. 
France now waved the mathematical sceptre. Mathematical 
studies among the English and German people had sunk to 
the lowest ebb. Among them the direction of original research 
was ill-chosen. The former adhered with excessive partiality 
to ancient geometrical methods ; the latter produced the com- 
binatorial school, which brought forth nothing of value. 

The labours of Euler, Lagrange, and Laplace lay in higher 
analysis, and this they developed to a wonderful degree. By 
them analysis came to be completely severed from geometry. 
During the preceding period the effort of mathematicians 
not only in England, but, to some extent, even on the conti- 
nent, had been directed toward the solution of problems 
clothed in geometric garb, and the results of calculation 
were usually reduced to geometric form. A change now 
took place. Euler brought about an emancipation of the 
analytical calculus from geometry and established it as an 
independent science. Lagrange and Laplace scrupulously 
adhered to this separation. Building on the broad foun- 
dation laid for higher analysis and mechanics by Newton 
and Leibniz, Euler, with matchless fertility of mind, erected 



EULBB, LAGRANGE, AND LAPLACE. 247 

an elaborate structure. There are few great ideas pursued 
by succeeding analysts which were not suggested by Euler, 
or of which he did not share the honour of invention. 
With, perhaps, less exuberance of invention, but with more 
comprehensive genius and profounder reasoning, Lagrange 
developed the infinitesimal calculus and placed analytical 
mechanics into the form in which we now know it. La- 
place applied the calculus and mechanics to the elaboration 
of the theory of universal gravitation, and thus, largely ex- 
tending and supplementing the labours of Newton, gave a full 
analytical discussion of the solar system. He also wrote an 
epoch-marking work on Probability. Among the analytical 
branches created during this period are the calculus of Varia- 
tions by Euler and Lagrange, Spherical Harmonics by La- 
place and Legendre, and Elliptic Integrals by Legendre. 

Comparing the growth of analysis at this time with the 
growth during the time of Gauss, Cauchy, and recent mathe- 
maticians, we observe an important difference. During the 
former period we witness mainly a development with refer- 
ence to form. Placing almost implicit confidence in results of 
calculation, mathematicians did not always pause to discover 
rigorous proofs, and were thus led to general propositions, 
some of which have since been found to be true in only special 
cases. The Combinatorial School in Germany carried this 
tendency to the greatest extreme ; they worshipped formalism 
and paid no attention to the actual contents of formulae. But 
in recent times there has been added to the dexterity in the 
formal treatment of problems, a much-needed rigour of demon- 
stration. A good example of this increased rigour is seen in 
the present use of infinite series as compared to that of Euler, 
and of Lagrange in his earlier works. 

The ostracism of geometry, brought about by the master- 
minds of this period, could not last permanently. Indeed, a 



248 A HISTORY OF MATHEMATICS. 

new geometric school sprang into existence in France before 
the close of this period. Lagrange would not permit a single 
diagram to appear in his M4canique analytique, but thirteen 
years before his death, Monge published his epoch-making 
Oiometrie descriptive, 

Leonhard Euler (1707-1783) was bom in Basel. His father, 
a minister, gave him his first instruction in mathematics and 
then sent him to the University of Basel, where he became a 
favourite pupil of John Bernoulli. In his nineteenth year he 
composed a dissertation on the masting of ships, which re- 
ceived the second prize from the French Academy of Sciences. 
When John Bernoulli's two sons, Daniel and Nicolaus, went to 
Russia, they induced Catharine I., in 1727, to invite their friend 
Euler to St. Petersburg, where Daniel, in 1733, was assigned to 
the chair of mathematics. In 1735 the solving of an astrono- 
mical problem, proposed by the Academy, for which several 
eminent mathematicians had demanded some months' time, 
was achieved in three days by Euler with aid of improved 
methods of his own. But the effort threw him into a fever 
and deprived him of the use of his right eye. With still 
superior methods this same problem was solved later by the 
illustrious Gauss in one hour ! ^ The despotism of Anne I. 
caused the gentle Euler to shrink from public affairs and to 
devote all his time to science. After his call to Berlin by 
Frederick the Great in 1747, the queen of Prussia, who 
received him kindly, wondered how so distinguished a scholar 
should be so timid and reticent. Euler naively replied, 
" Madam, it is because I come from a country where, when one 
speaks, one is hanged." In 1766 he with diflSculty obtained 
permission to depart from Berlin to accept a call by Catha- 
rine II. to St. Petersburg. Soon after his return to Russia he 
became blind, but this did not stop his wonderful literary 
productiveness, which continued for seventeen years, until the 



BULER, LAGRANGE, AND LAPLACE. 249 

day of his death .^ He dictated to his servant his Anleitung 
zur Algebra^ 1770, which, though purely elementary, is meri- 
torious as one of the earliest attempts to put the fundamental 
processes on a sound basis. 

Euler wrote an immense number of works, chief of which 
are the following: Introductio in ancUysin infinitorum, 1748, 
a work that caused a revolution in analytical mathematics, a 
subject which had hitherto never been presented in so general 
and systematic manner ; InstiitUiones calculi differenlialiSy 1755, 
and InatUutiones calculi integralis, 1768-1770, which were the 
most complete and accurate works on the calculus of that time, 
and contained not only a full summary of everything then 
known on this subject, but also the Beta and Gamma Func- 
tions and other original investigations; Methodxis inveniendi 
lineas curvas nuiximi minimive proprietate gaudentes, 1744, 
which, displaying an amount of mathematical genius seldom 
rivalled, contained his researches on the calculus of variations 
(a subject afterwards improved by Lagrange), to the invention 
of which Euler was led by the study of isoperimetrical curves, 
the brachistochrone in a resisting medium, and the theory of 
geodesies (subjects which had previously engaged the attention 
of the elder BernouUis and others) ; the Theoria motuum plane- 
tarum et cometarum, 1744, Theoria motus lunacy 1753, Theoria 
motuum lunte, 1772, are his chief works on astronomy; Ses 
lettres d une princesse d^Allemagne mr quelques sujets de 
Physique et de Philosophie, 1770, was a work which enjoyed 
great popularity. 

We proceed to mention the principal innovations and inven- 
tions of Euler. He treated trigonometry as a branch of 
analysis, introduced (simultaneously with Thomas Simpson in 
England) the now current abbreviations for trigonometric 
functions, and simplified formulae by the simple expedient 
of designating the angles of a triangle by A, B, C, and the 



250 A HISTORY OF HATHSBCATICS. 

opposite sides by a, by c, respectively. He pointed out the 
relation between trigonometric and exponential functions. In 
a paper of 1737 we first meet the symbol ir to denote 3.14159«-.*^ 
Euler laid down the rules for the transformation of co-ordinates 
in space, gave a methodic analytic treatment of plane curves 
and of surfaces of the second order. He was the first to 
discuss the equation of the second degree in three variables, 
and to classify the surfaces represented by it. By criteria 
analogous to those used in the classification of conies he 
obtained five species. He devised a method of solving bi- 
quadratic equations by assuming x = Vp -f -y/q -f Vr, with the 
hope that it would lead him to a general solution of algebraic 
equations. The method of elimination by solving a series of 
linear equations (invented independently by Bezout) and the 
method of elimination by symmetric functions, are due to him.*'^ 
Far reaching are Euler's researches on logarithms. Leibniz 
and John Bernoulli once argued the question whether a 
negative number has a logarithm. Bernoulli claimed that 
since ( — a)* = (-f a)', we have log(— a)* = log(-f a)* and 
2 log (—a) = 2 log (-f- a), and finally log ( — a) = log (-f a). 
Euler proved that a has really an infinite number of loga- 
rithms, all of which are imaginary when a is negative, and all 
except one when a is positive. He then explained how 
log (—a)* might equal log(-f-a)^, and yet log (—a) not 
equal log (-fa). 

The subject of infinite series received new life from him. 
To his researches on series we owe the creation of the theory of 
definite integrals by the development of the so-called Eulerian 
integrals. He warns his readers occasionally against the use 
of divergent series, but is nevertheless very careless himself. 
The rigid treatment to which infinite series are subjected now 
was then undreamed of. No clear notions existed as to what 
constitutes a convergent series. Neither Leibniz nor Jacob 



EULEK, LAGRANGE, AND LAPLACE. 251 

and John Bernoulli had entertained any serious doubt of the 
oorrectness of the expression ^ = 1 — 1 + 1 — ! + ••.. Guido 
Grand! went so far as to conclude from this that ^ = + + 

+ •••. In the treatment of series Leibniz advanced a meta- 
physical method of proof which held sway over the minds of 
the elder Bemoullis, and even of Euler.** The tendency of 
that reasoning was to justify results which seem to us now 
highly absurd. The looseness of treatment can best be seen 
from examples. The very paper in which Euler cautions 
against divergent series contains the proof that 

•••— H h 1 + » + n*-f ••• = as follows: 

IV n 

n-|-n'+... =- , IH \-—A = -J 

1 — n n nr n — 1 

these added give zero. Euler has no hesitation to write 

1 — 3 + 5 — 7 H — =0, and no one objected to such results 
excepting Nicolaus Bernoulli, the nephew of John and Jacob. 
Strange to say, Euler finally succeeded in converting Nicolaus 
Bernoulli to his own erroneous views. At the present time 
it is difficult to believe that Euler should have confidently 
written sin (^ — 2 sin 2 (^ + 3 sin 3 (^ — 4 sin 4 <^ + ••• = 0, but 
such examples afford striking illustrations of the want of 
scientific basis of certain parts of analysis at that time. 
Euler's proof of the binomial formula for negative and 
fractional exponents, which has been reproduced in elemen- 
tary text-books of even recent years, is faulty. A remarkable 
development, due to Euler, is what he named the hypergeo- 
metric series, the summation of which he observed to be 
dependent upon the integration of a linear differential equa- 
tion of the second order, but it remained for Gauss to point 
out that for special values of its letters, this series represented 
nearly all functions then known. 

Euler developed the calculus of finite differences in the first 



252 A HISTORY OF MATHEMATICS. 

chapters of his InstUutiones calculi differentialiSy and then 
deduced the differential calculus from it. He established a 
theorem on homogeneous functions, known by his name, and 
contributed largely to the theory of differential equations, a 
subject which had received the attention of Newton, Leibniz, 
and the Bernoullis, but was still undeveloped. Clairaut, 
Fontaine, and Euler about the same time observed criteria of 
integrability, but Euler in addition showed how to employ 
them to determine integrating factors. The principles on 
which the criteria rested involved some degree of obscurity. 
The celebrated addition-theorem for elliptic integrals was first 
established by Euler. He invented a new algorithm for 
continued fractions, which he employed in the solution of 
the indeterminate equation oa: -f 6y = c. We now know that 
substantially the same solution of this equation was given 
1000 years earlier, by the Hindoos. By giving the factors of 
the number 2^* -f 1 when n==6j he pointed out that this ex- 
pression did not always represent primes, as was supposed by 
Fermat. He first supplied the proof to "Fermat's theorem," 
and to a second theorem of Fermat, which states that every 
prime of the form 4n-f 1 is expressible as the sum of two 
squares in one and only one way. A third theorem of Fermat, 
that 7f -\-y'' =■ z", has no integral solution for values of n 
greater than 2, was proved by Euler to be correct when n = 3. 
Euler discovered four theorems which taken together make 
out the great law of quadratic reciprocity, a law independently 
discovered by Legendre.*^ Euler enunciated and proved a 
well-known theorem, giving the relation between the number 
of vertices, faces, and edges of certain polyhedra, which, 
however, appears to have been known to Descartes. The 
powers of Euler were directed also towards the fascinating 
subject of the theory of probability, in which he solved some 
difficult problems. 



EULBB, LAGRANGE, AND LAPLACE. 258 

Of no little importance are Euler's labours in analytical 
^nechanics. Says Whewell: "The person who did most to 
^ve to analysis the generality and symmetry which are now 
"Sts pride, was also the person who made mechanics analytical ; 
3 mean Euler.''" He worked out the theory of the rotation of 
9 body around a fixed point, established the general equations 
of motion of a free body, and the general equation of hydrody- 
namics. He solved an immense number and variety of mechan- 
ical problems, which arose in his mind on all occasions. Thus, 
on reading Virgil's lines, " The anchor drops, the rushing keel 
is staid," he could not help inquiring what would be the 
ship's motion in such a case. About the same time as Daniel 
Bernoulli he published the Principle of the Conservation of 
Areas and defended the principle of " least action," advanced 
by Maupertius. He wrote also on tides and on sound. 

Astronomy owes to Euler the method of the variation of 
arbitrary constants. By it he attacked the problem of per- 
turbations, explaining, in case of two planets, the secular vari- 
ations of eccentricities, nodes, etc. He was one of the first 
to take up with success the theory of the moon's motion by 
giving approximate solutions to the " problem of three bodies." 
He laid a sound basis for the calculation of tables of the moon. 
These researches on the moon's motion, which captured two 
prizes, were carried on while he was blind, with the assistance 
of his sons and two of his pupils. 

Most of his memoirs are contained in the transactions of 
the Academy of Sciences at St. Petersburg, and in those of 
the Academy at Berlin. From 1728 to 1783 a large portion 
of the Petropolitan transactions were filled by his writings. 
He had engaged to furnish the Petersburg Academy with 
memoirs in sufficient number to enrich its acts for twenty 
years — a promise more than fulfilled, for down to 1818 the 
Yoliimes usually contained one or more papers of his. It has 



264 A HISTORY OF HATHSMATICS. 

been said that an edition of Euler's complete works would fill 
16,000 quarto pages. His mode of working was, first to con- 
centrate his powers upon a special problem, then to solve 
separately all problems growing out of the first No one 
excelled him in dexterity of accommodating methods to special 
problems. It is easy to see that mathematicians could not 
long continue in Euler's habit of writing and publishing. The 
material would soon grow to such enormous proportions as to 
be unmanageable. We are not surprised to see almost the 
opposite in Lagrange, his great successor. The great French- 
man delighted in the general and abstract, rather than, like 
Euler, in the special and concrete. His writings are con- 
densed and give in a nutshell what Euler narrates at great 
length. 

Jean-le-Rond D'Alembert (1717-1783) was exposed, when 
an infant, by his mother in a market by the church of St. 
Jean-le-Kond, near the N6tre-Dame in Paris, from which he 
derived his Christian name. He was brougM up by the wife 
of a poor glazier. It is said that when he began to show signs 
of great talent, his mother sent for him, but received the 
reply, "You are only my step-mother; the glazier's wife is 
my mother." His father provided him with a yearly income. 
D'Alembert entered upon the study of law, but such was his 
love for mathematics, that law was soon abandoned. At the 
age of twenty-four his reputation as a mathematician secured 
for him admission to the Academy of Sciences. In 1743 
appeared his TraitS de dynamique, founded upon the important 
general principle bearing his name : The impressed forces are 
equivalent to the effective forces. D'Alembert's principle 
seems to have been recognised before him by Fontaine, and 
in some measure by John Bernoulli and Newton. D'Alembert 
gave it a clear mathematical form and made numerous appli- 
cations of it. It enabled the laws of motion and the reason- 



BULEB, LAGBANGB, AND LAPLAOB. 265 

ings depending on them to be represented in the most general 
form, in analytical language. D'Alembert applied it in 1744 
in a treatise on the equilibrium and motion of fluids, in 1746 
to a treatise on the general causes of winds, which obtained 
a prize from the Berlin Academy. In both these treatises, as 
also in one of 1747, discussing the famous problem of vibrating 
chords, he was led to partial differential equations. He was 

a leader among the pioneers in the study of such equations. 

dhi Shi 
To the equation ^ = a' -^ arising in the problem of vibrat- 
ing chords, he gave as the general solution, 

and showed that there is only one arbitrary function, if y be 
supposed to vanish for 2b = and 2b = /. Daniel Bernoulli, 
starting with a particular integral given by Brook Taylor, 
showed that this differential equation is satisfied by the 
trigonometric series 

y=asin^.cos^-h/8sin^.cos^' + ..., 

and claimed this expression to be the most general solution. 
Euler denied its generality, on the ground that, if true, the 
doubtful conclusion would follow that the above series repre- 
sents any arbitrary function of a variable. These doubts were 
dispelled by Fourier. Lagrange proceeded to find the sum 
of the above series, but D'Alembert rightly objected to his 
process, on the ground that it involved divergent series.** 

A most beautiful result reached by lyAlembert, with aid 
of his principle, was the complete solution of the problem of 
the precession of the equinoxes, which had baffled the talents 
of the best minds. He sent to the French Academy in 1747, 
on the same day with Clairaut, a solution of the problem of 
three bodies. This had become a question of universal inter- 



256 A HISTORY OF MATHEMATICS. 

est to mathematicians, in which each vied to outdo all others. 
The problem of two bodies, requiring the determination of 
their motion when they attract each other with forces in- 
versely proportional to the square of the distance between 
them, had been completely solved by Newton. The "problem 
of three bodies " asks for the motion of three bodies attracting 
each other according to the law of gravitation. Thus far, 
the complete solution of this has transcended the power of 
analysis. The general differential equations of motion were 
stated by Laplace, but the difficulty arises in their integration. 
The "solutions" hitherto given are merely convenient methods 
of approximation in special cases when one body is the sun, 
disturbing the motion of the moon around the earth, or where 
a planet moves under the influence of the sun and another 
planet. 

In the discussion of the meaning of negative quantities, of 
the fundamental processes of the calculus, and of the theory of 
probability, D'Alembert paid some attention to the philosophy 
of mathematics. His criticisms were not always happy. In 
1754 he was made permanent secretary of the French Academy. 
During the last years of his life he was mainly occupied with 
the great French encyclopaedia, which was begun by Diderot 
and himself. D'Alembert declined, in 1762, an invitation of 
Catharine II. to undertake the education of her son. Frederick 
the Great pressed him to go to Berlin. He made a visit, but 
declined a permanent residence there. 

Alexis Claude Clairaut (1713-1765) was a youthful prodigy. 
He read THospital's works on the infinitesimal calculus and on 
conic sections at the age of ten. In 1731 was published his 
Recherches sur lea courbes d double courbure, which he had ready 
for the press when he was sixteen. It was a work of remark- 
able elegance and secured his admission to the Academy of 
Sciences when still under legal age. In 1731 he gave a proof of 



BDI^ER, LAOBANQB, AND LAPLAOK. 



25T 



the theorem enunciated by Xevton, that every oubio is « pn>> 
jection of one of five divergent parabolas. Clairaut formed the 
acquaintance of Maupertius, whom he accompanied on an expe- 
dition to Lapland to measure the length of a degree of the 
meridian. At that time the shape of the earth was a subject 
of serious disagreement. Newton and Huygens bad concluded 
from theory that the earth was flattened at the poles. About 
1713 Dominico Cassini measured an arc extending from Dunkirk 
to Perpignan and arrived at the startling result that the earth 
IB elongated at the poles. To decide between the conflicting 
opinions, measurements were renewed. Maupertius earned by 
his work in Lapland the title of " earth flattener " by dispiov- 
ing the Cassinian tenet that the earth was elongated at the 
poles, and showing that Newton was right. On his return, in 
1743, Clairaut published a work, Th4orie de la figure de la Terrty 
which was based on the results of Maclaurin on homogeneous 
ellipsoids. It contains a remarkable theorem, named after 
Clairaut, that the sum of the fractions expressing the ellipticity 
and the increase of gravity at the pole is equal to 2\ times the 
fraction expressing the centrifugal force at the equator, the 
unit of force being represented \tj the force of gravity at the 
equator. This theorem is indepeadwtt of any hypollicjiis with 
respect to the law of densititi Sr^' iawowige strata of tlic 
earth. It embodies most of CbiruiL'ii r^n-virahw. Todhuiitrr 
says that "in the tignre "f ■ i '--j^'' ■" nth^j^nHin has 
accomplished ho muah as CliJ< ' 

present Bubnt^aitiallv 1= 'if i<'f" -_■-..'■ ' -™ U diffannL 
The splendid iujalyabi "' ' ^- <1 butdld 

. really altas ftb» a,' 

indB of C 

kin 1753 >- 






i 



258 A HISTOBT OF MATHEMATICS. 

the explanation of the motion of the lunar apsides. This 
motion, left unexplained by Newton, seemed to him at first 
inexplicable by Newton's law, and he was on the point of 
advancing a new hypothesis regarding gravitation, when, tak- 
ing the precaution to carry his calculation to a higher degree 
of approximation, he reached results agreeing with observa- 
tion. The motion of the moon was studied about the same 
time by Euler and I^Alembert. Clairaut predicted that 
"Halley's Comet," then expected to return, would arrive at 
its nearest point to the sim on April 13, 1759, a date which 
turned out to be one month too late. He was the first to 
detect singular solutions in differential equations of the first 
order but of higher degree than the first. 

In their scientific labours there was between Clairaut and 
lyAlembert great rivalry, often far from friendly. The grow- 
ing ambition of Clairaut to shine in society, where he was a 
great favourite, hindered his scientific work in the latter part 
of his life. 

Johann Heinrich Lambert (1728-1777), born at Miihlhausen 
in Alsace, was the son of a poor tailor. While working at his 
father's trade, he acquired through his own unaided efforts a 
knowledge of elementary mathematics. At the age of thirty he 
became tutor in a Swiss family and secured leisure to continue 
his studies. In his travels with his pupils through Europe he 
became acquainted with the leading mathematicians. In 1764 
he settled in Berlin, where he became member of the Academy, 
and enjoyed the society of Euler and Lagrange. He received 
a small pension, and later became editor of the Berlin Ephem- 
eris. His many-sided scholarship reminds one of Leibniz. 
In his CosmologiccU Letters he made some remarkable prophe- 
cies regarding the stellar system. In mathematics he made 
several discoveries which were extended and overshadowed by 
his great contemporaries. His first research on pure mathe- 



EULBR, LAORAN6B, AND LAPLAOB. 259 

matics developed in an infinite series the root x of the equation 
^-i-px^q. Since each equation of the form oaf -{-hof^d 
can be reduced to af +pa5 = g in two ways, one or the other of 
the two resulting series was always found to be convergent, 
and to give a value of x, Lambert's results stimulated Euler, 
who extended the method to an equation of four terms, and 
particularly Lagrange, who found that a function of a root of 
a — x+(^(a;) = can be expressed by the series bearing his 
name. In 1761 Lambert communicated to the Berlin Academy 
a memoir, in which he proves that ir is irrational. This proof 
is given in Note IV. of Legendre's 04ometrie, where it is 
extended to ir*. To the genius of Lambert we owe the intro- 
duction into trigonometry of hyperbolic functions, which he 
designated by ainh x, cosh x, etc. His Freye Perspective, 1759 
and 1773, contains researches on descriptive geometry, and 
entitle him to the honour of being the forerunner of Monge. 
In his effort to simplify the calculation of cometary orbits, he 
was led geometrically to some remarkable theorems on conies, 
for instance this : '< If in two ellipses having a common major 
axis we take two such arcs that their chords are equal, and 
that also the sums of the radii vectores, drawn respectively 
from the foci to the extremities of these arcs, are equal to 
each other, then the sectors formed in each ellipse by the arc 
and the two radii vectores are to each other as the square 
roots of the parameters of the ellipses." ^ 

John Landen (1719-1790) was an English mathematician 
whose writings served as the starting-point of investigations 
by Euler, Lagrange, and Legendre. Landen's capital discov- 
ery, contained in a memoir of 1755, was that every arc of the 
hyperbola is immediately rectified by means of two arcs of an 
ellipse. In his "residual analysis" he attempted to obviate 
the metaphysical difficulties of fluxions by adopting a purely 
algebraic method. Lagrange's CcUaU des Fonctiona is based 



260 A HISTORY OF MATHEMATICS. 

upon this idea. Landen showed how the algebraic expression 
for the roots of a cubic equation could be derived by applica- 
tion of the differential and integral calculus. Most of the 
time of this suggestive writer was spent in the pursuits of 
active life. 

Etienne B^zout (1730-1783) was a French writer of popular 
mathematical school-books. In his Thdorie g6n4rale des Equor 
lions Alg^briqms, 1779, he gave the method of elimination by 
linear equations (invented also by Euler). This method was 
first published by him in a memoir of 1764, in which he uses 
determinants, without, however, entering upon their theory. 
A beautiful theorem as to the degree of the resultant goes by 
his name. 

Louis Arbogaste •^1759-1803) of Alsace was professor of 
mathematics at Strasburg. His chief work, the Calcul des 
Dirivations^ 1800^ gives the method known by his name, by 
which the successive coefficients of a development are derived 
from one another when the expression is complicated. De 
Morgan has pointed out that the true nature of derivation 
is differentiation accompanied by integration. In this book 

for the first time are the symbols of operation separated from 

dv 
those of quantity. The notation D^ for -^ is due to him. 

Maria Gaetana Agnesi (1718-1799) of Milan, distinguished as 
a linguist, mathematician, and philosopher, filled the mathe- 
matical chair at the University of Bologna during her father's 
sickness. In 1748 she published her Instituzioni AnalUiche, 
which was translated into English in 1801. The "witch of 
Agnesi " or " versiera " is a plane curve containing a straight 

line, 05 = 0, and a cubic ( - ) -f 1 = - . 

\cj X 

Joseph Louis Lagrange (1736-1813), one of the greatest 
mathematicians of all times, was born at Turin and died at 
Paris. He was of French extraction. His father, who had 



EULER, LAGRANGE, AND LAPLACE. 261 

charge of the Sardinian military chest, was once wealthy, but 
lost all he had in speculation. Lagrange considered this loss 
his good fortune, for otherwise he might not have made math- 
ematics the pursuit of his life. While at the college in Turin 
his genius did not at once take its true bent. Cicero and Vir- 
gil at first attracted him more than Archimedes and Newton. 
He soon came to admire the geometry of the ancients, but the 
perusal of a tract of Halley roused his enthusiasm for the 
analytical method, in the development of which he was des- 
tined to reap undying glory. He now applied himself to 
mathematics, and in his seventeenth year he became professor 
of mathematics in the royal military academy at Turin. 
Without assistance or guidance he entered upon a course of 
study which in two years placed him on a level with the 
greatest of his contemporaries. With aid of his pupils he 
established a society which subsequently developed into the 
Turin Academy. In the first five volumes of its transactions 
appear most of his earlier papers. At the age of nineteen he 
communicated to Euler a general method of dealing with 
<' isoperimetrical problems," known now as the Calculus of 
Variations. This commanded Euler's lively admiration, and 
he courteously withheld for a time from publication some 
researches of his own on this subject, so that the youthful 
Lagrange might complete his investigations and claim the 
invention. Lagrange did quite as much as Euler towards the 
creation of the Calculus of Variations. As it came from Euler 
it lacked an analytic foundation, and this Lagrange supplied. 
He separated the principles of this calculus from geometric 
considerations by which his predecessor had derived them. 
Euler had assumed as fixed the limits of the integral, t.6. the 
extremities of the curve to be determined, but Lagrange 
removed this restriction and allowed all co-ordinates of the 
curve to vary at the same time. Euler introduced in 1766 the 



262 A HISTORY OF MATHBMATIGS. 

name '^ calculus of variations/' and did much to improve this 
science along the lines marked out by Lagrange. 

Another subject engaging the attention of Lagrange at 
Turin was the propagation of sound. In his papers on this 
subject in the Miscellanea Taurinensia, the young mathemati- 
cian appears as the critic of Newton, and the arbiter between 
Euler and D'Alembert. By considering only the particles 
which are in a straight line, he reduced the problem to the 
same partial differential equation that represents the motions 
of vibrating strings. The general integral of this was found by 
D'Alembert to contain two arbitrary functions, and the ques- 
tion now came to be discussed whether an arbitrary function 
may be discontinuous. D'Alembert maintained the negative 
against Euler, Daniel Bernoulli, and finally Lagrange, — argu- 
ing that in order to determine the position of a point of the 
chord at a time t, the initial position of the chord must be 
continuous. Lagrange settled the question in the affirmative. 

By constant application during nine years, Lagrange, at the 
age of twenty-six, stood at the summit of European fame. 
But his intense studies had seriously weakened a constitution 
never robust, and though his physicians induced him to take 
rest and exercise, his nervous system never fully recovered its 
tone, and he was thenceforth subject to fits of melancholy. 

In 1764 the French Academy proposed as the subject of 
a prize the theory of the libration of the moon. It demanded 
an explanation, on the principle of universal gravitation, why 
the moon always turns, with but slight variations, the same 
phase to the earth. Lagrange secured the prize. This suc- 
cess encouraged the Academy to propose as a prize the theory 
of the four satellites of Jupiter, — a problem of six bodies, 
more difficult than the one of three bodies previously solved 
by Clairaut, D*Alembert, and Euler. Lagrange overcame the 
difficulties, but the shortness of time did not permit him to 



EXH/BB, LAGRANGE, AND LAPLACE. 263 

exhaust the subject. Twenty-four years afterwards it was 
completed by Laplace. Later astronomical investigations of 
Lagrange are on cometary perturbations (1778 and 1783), on 
Kepler's problem, and on a new method of solving the prob- 
lem of three bodies. 

Being anxious to make the personal acquaintance of leading 
mathematicians, Lagrange visited Paris, where he enjoyed the 
stimulating delight of conversing with Clairaut, D'Alembert, 
Condorcet, the Abb^ Marie, and others. He had planned a 
visit to London, but he fell dangerously ill after a dinner in 
Paris, and was compelled to return to Turin. In 1766 Euler 
left Berlin for St. Petersburg, and he pointed out Lagrange as 
the only man capable of filling the place. I^Alembert recom- 
mended him at the same time. Frederick the Great there- 
upon sent a message to Turin, expressing the wish of ''the 
greatest king of Europe '^ to have '' the greatest mathemati- 
cian " at his court. Lagrange went to Berlin, and staid there 
twenty years. Finding all his colleagues married, and being 
assured by their wives that the marital state alone is happy, 
he married. The union was not a happy one. His wife 
soon died. Frederick the Great held him in high esteem, 
and frequently conversed with him on the advantages of per- 
fect regularity of life. This led Lagrange to cultivate regular 
habits. He worked no longer each day than experience taught 
him he could without breaking down. His papers were care- 
fully thought out before he began writing, and when he wrote 
he did so without a single correction. 

During the twenty years in Berlin he crowded the transac- 
tions of the Berlin Academy with memoirs, and wrote also 
the epoch-making work called the M4canique Analytique, He 
enriched algebra by researches on the solution of equations. 
There are two methods of solving directly algebraic equa- 
tions, — that of substitution and that of combination. The 



264 A HI8T0BY OF MATHEMATICS. 

former method was developed by Ferrari, Vieta, Tchimhausen, 
Euler, B^zout, and Lagrange ; the latter by Yandermonde and 
Lagrange.''^ In the method of substitution the original forms 
are so transformed that the determination of the roots is made 
to depend upon simpler functions (resolvents). In the method 
of combination auxiliary quantities are substituted for certain 
simple combinations ("types") of the unknown roots of the 
equation, and auxiliary equations (resolvents) are obtained for 
these quantities with aid of the coefficients of the given equa- 
tion. Lagrange traced all known algebraic solutions of equa- 
tions to the uniform principle consisting in the formation and 
solution of equations of lower degree whose roots are linear 
functions of the required roots, and of the roots of unity. He 
showed that the quintic cannot be reduced in this way, its 
resolvent being of the sixth degree. His researches on the 
theory of equations were continued after he left Berlin. In 
the R48olution des Equations numhiques (1798) he gave a 
method of approximating to the real roots of numerical equa- 
tions by continued fractions. Among other things, it contains 
also a proof that every equation must have a root, — a theorem 
which appears before this to have been considered self-evident. 
Other proofs of this were given by Argand, Gauss, and Cauchy. 
In a note to the above work Lagrange uses Fermat's theorem 
and certain suggestions of Gauss in effecting a complete alge- 
braic solution of any binomial equation. 

While in Berlin Lagrange published several papers on the 
theory of numbers. In 1769 he gave a solution in integers of 
indeterminate equations of the second degree, which resembles 
the Hindoo cyclic method ; he was the first to prove, in 1771, 
"Wilson's theorem," enunciated by an Englishman, John 
Wilson, and first published by Waring in his Meditationes 
Algebraicoe; he investigated in 1775 under what conditions 
± 2 and ± 5 (— 1 and ± 3 having been discussed by Euler) 



EULER, LAGKANOE, AND LAPLACE. 265 

are quadratic residues, or non-residues of odd prime numbers, 
q ; he proved in 1770 M^ziriac*s theorem that every integer is 
equal to the sum of four, or a less number, of squares. He 
proved Format's theorem on af -f y* = «*, for the case n ss 4, 
also Fermat's theorem that, if a' + &' = c', then ab is not a 
square. 

In his memoir on Pyramids, 1773, Lagrange made consider- 
able use of determinants of the third order, and demonstrated 
that the square of a determinant is itself a determinant. He 
never, however, dealt explicitly and directly with determi- 
nants; he simply obtained accidentally identities which are 
now recognised as relations between determinants. 

Lagrange wrote much on differential equations. Though 
the subject of contemplation by the greatest mathematicians 
(Euler, I^Alembert, Clairaut, Lagrange, Laplace), yet more 
than other branches of mathematics did they resist the sys- 
tematic application of fixed methods and principles. Lagrange 
established criteria for singular solutions {CcUctd dea Fonctians, 
Lessons 14-17), which are, however, erroneous. He was the 
first to point out the geometrical significance of such solutions. 
He generalised Euler's researches on total differential equa- 
tions of two variables, and of the ninth order ; he gave a solu- 
tion of partial differential equations of the first order {Berlin 
Memoirs, 1772 and 1774), and spoke of their singular solutions, 
extending their solution in Memoirs of 1779 and 1785 to equa- 
tions of any number of variables. The discussion on partial 
differential equations of the second order, carried on by 
lyAlembert, Euler, and Lagrange, has already been referred 
to in our account of I^Alembert. 

While in Berlin, Lagrange wrote the ^^Mickanique Analytique," 
the greatest of his works (Paris, 1788). From the principle 
of virtual velocities he deduced, with aid of the calculus of 
variations, the whole system of mechanics so elegantly and 



266 A HISTOKY OF MATHEMATICS. 

harmoniously that it may fitly be called^ in Sir William 
Rowan Hamilton's words, '' a kind of scientific poem." It is a 
most consummate example of analytic generality. Greometrical 
figures are nowhere allowed. '^ On ne trouvera point de figures 
dans cet ouvrage " (Preface). The two divisions of mechanics 
— statics and dynamics — are in the first four sections of each 
carried out analogously, and each is prefaced by a historic 
sketch of principles. Lagrange formulated the principle of 
least action. In their original form, the equations of motion 
involve the co-ordinates x, y, z, of the different particles m or 
dm of the system. But x, y, z, are in general not independent, 
and Lagrange introduced in place of them any variables (, ^, 
^, whatever, determining the position of the point at the time. 
These may be taken to be independent. The equations of 
motion may now assume the form 

ddT dT , ^ r. 
dtdi* di^ ' 

or when B, ^, ^, . . . are the partial differential coefficients 
with respect to ^, ^, ^, . . . of one and the same function V, 

then the form 

d^dT__dT dV^Q 
dt d$' di di 

The latter is par excellence the Lagrangian form of the equa- 
tions of motion. With Lagrange originated the remark that 
mechanics may be regarded as a geometry of four dimensions. 
To him falls the honour of the introduction of the potential 
into dynamics.^ Lagrange was anxious to have his M^canique 
Analytique published in Paris. The work was ready for print 
in 1786, but not till 1788 could he find a publisher, and then 
only with the condition that after a few years he would pur- 
chase all the unsold copies. The work was edited by 
Legendre. 



BULER, LAOBANGE, AND LAPLACB. 267 

After the death of Frederick the Great, men of science 
were no longer respected in Germany, and Lagrange accepted 
an invitation of Louis XVI. to migrate to Paris. The French 
queen treated him with regard, and lodging was procured for 
him in the Louvre. But he was seized with a long attack of 
melancholy which destroyed his taste for mathematics. For 
two years his printed copy of the M4canique, fresh from the 
press, — the work of a quarter of a century, — lay unopened on 
his desk. Through Lavoisier he became interested in chem- 
istry, which he found " as easy as algebra." The disastrous 
crisis of the French Revolution aroused him again to activity. 
About this time the young and accomplished daughter of the 
astronomer Lemonnier took compassion on the sad, lonely 
Lagrange, and insisted upon marrying him. Her devotion to 
him constituted the one tie to life which at the approach of 
death he found it hard to break. 

He was made one of the commissioners to establish weights 
and measures having units founded on nature. Lagrange 
strongly favoured the decimal subdivision, the general idea of 
which was obtained from a work of Thomas Williams, London, 
1788. Such was the moderation of Lagrange's character, and 
such the universal respect for him, that he was retained as presi- 
dent of the commission on weights and measures even after it 
had been purified by the Jacobins by striking out the names 
of Lavoisier, Laplace, and others. Lagrange took alarm at the 
fate of Lavoisier, and planned to return to Berlin, but at the 
establishment of the Ecole Normale in 1795 in Paris, he was 
induced to accept a professorship. Scarcely had he time to 
elucidate the foundations of arithmetic and algebra to young 
pupils, when the school was closed. His additions to the 
alpfcbra of Euler were prepared at this time. In 1797 the 
Ecole Polytechnique was founded, with Lagrange as one of 
the professors. The earliest triumph of this institution was 



/• 



268 A HISTORY OF MATHEMATICS. 

the restoration of Lagrange to analysis. His mathematical 
activity burst out anew. He brought forth the Th^arie dea 
fonctions analytiques (1797), Legons axir le calcul dea foncHona, 
a treatise on the same lines as the preceding (1801), BJ^d the 
R^olution dea ^quationa nurtieriquea (1798). In 1810 he 
began a thorough revision of his Micanique ancUytique, but 
he died before its completion. 

The Thiorie dea fonctiana, the germ of which is found in a 
memoir of his of 1772, aimed to place the principles of the 
calculus upon a sound foundation by relieving the mind of the 
difficult coDception of a limit or infinitesimal. John Landen's 
residual calculus, professing a similar object, was unknown to 
him. Lagrange attempted to prove Taylor's theorem (the 
power of which he was the first to point out) by simple algebra, 
and then to develop the entire calculus from that theorem. 
The principles of the calculus were in his day involved in 
philosophic difficulties of a serious nature. The infinitesimals 
of Leibniz had no satisfactory metaphysical basis. In the 
differential calculus of Euler they were treated as absolute 
zeros. In Newton's limiting ratio, the magnitudes of which it 
is the ratio cannot be found, for at the moment when they 
should be caught and equated, there is neither arc nor chord. 
The chord and arc were not taken by Newton as equal before 
vanishing, nor after vanishing, but when they vanish. " That 
method,'' said Lagrange, " has the great inconvenience of con- 
sidering quantities in the state in which they cease, so to 
speak, to be quantities ; for though we can always well con- 
ceive the ratios of two quantities, as long as they remain 
finite, that ratio offers to the mind no clear and precise idea, 
as soon as its terms become both nothing at the same time." 
D'Alembert's method of limits was much the same as the 
method of prime and ultimate ratios. D'Alembert taught 
that a variable actually reached its limit. When Lagrange 



EUL£B, LAGRANGB, AND LAPLACE. 269 

endeavoured to free the calculus of its metaphysical difficulties, | 
by resorting to common algebra, he avoided the whirlpool of i 
Charybdis only to suffer wreck against the rocks of Scylla. 
The algebra of his day, as handed down to him by Euler, was j 
founded on a false view of infinity. No correct theory of • 
infinite series had then been established. Lagrange proposed 
to define the differential coefficient of f(x) with respect to x 
as the coefficient of h in the expansion of f{x -f h) by Taylor's 
theorem, and thus to avoid all reference to limits. But he / 
used infinite series without ascertaining that they were con- 1 
vergent, and his proof that f{x -f h) can always be expanded/ 
in a series of ascending powers of h, labours under serious' . 
defects. Though Lagrange's method of developing the calculus 
was at first greatly applauded, its defects were fatal, and to-dai^ 
his " method of derivatives," as it was called, has been geij- 
erally abandoned. He introduced a notation of his own, bu,t 
it was inconvenient, and was abandoned by him in the second 
edition of his M^canique^ in which he used infinitesimals. The 
primary object of the TMorie des fonciiona was not attained, 
but its secondary results were far-reaching. It was a purely 
abstract mode of regarding functions, apart from geometrical 
or mechanical considerations. In the further development 
of higher analysis a function became the leading idea, and 
Lagrange's work may be regarded as the starting-point of the 
theory of functions as developed by Cauchy, Biemann, Weier- 
strass, and others. 

In the treatment of infinite series Lagrange displayed in 
his earlier writings that laxity common to all mathematicians 
of his time, excepting Nicolaus Bernoulli II. and D'Alembert 
But his later articles mark the beginning of a period of greater 
rigour. Thus, in the CcUcul de fonctions he gives his theorem 
on the limits of Taylor's theorem. Lagrange's mathematical 
researches extended to subjects which have not been men- 



270 A HISTORY OF MATHEMATICS. 

tioned here — such as probabilities^ finite differences, ascend- 
ing continued fractions, elliptic integrals. Everywhere his 
wonderful powers of generalisation and abstraction are made 
manifest. In that respect he stood without a peer, but 
his great contemporary, Laplace, surpassed him in practical 
sagacity. Lagrange was content to leave the application of 
his general results to others, and some of the most important 
researches of Laplace (particularly those on the velocity of 
sound and on the secular acceleration of the moon) are im- 
plicitly contained in Lagrange's works. 

Lagrange was an extremely modest man, eager to avoid 
controversy, and even timid in conversation. He spoke in 
tones of doubt, and his first words generally were, " Je ne 
sais pas." He would never allow his portrait to be taken, 
and the only ones that were secured were sketched without 
his knowledge by persons attending the meetings of the 
Institute. 

Pierre Simon Laplace (1749-1827) was born at Beaumont- 
en-Auge in Normandy. Very little is known of his early 
life. When at the height of his fame he was loath to speak 
of his boyhood, spent in poverty. His father was a small 
farmer. Some rich neighbours who recognised the boy's 
talent assisted him in securing an education. As an extern 
he attended the military school in Beaumont, where at an 
early age he became teacher of mathematics. At eighteen 
he went to Paris, armed with letters of recommendation to 
D'Alembert, who was then at the height of his fame. The 
letters remained unnoticed, but young Laplace, undaunted, 
wrote the great geometer a letter on the principles of me- 
chanics, which brought the following enthusiastic response : 
" You needed no introduction ; you have recommended your- 
self ; my support is your due." D'Alembert secured him a 
position at the EcoU Militaire of Paris as professor of raathe- 



BULEB, LAGRANGE, AND LAPLACB. 271 

matics. His future was now assuredi and he entered upon 
those profound researches which brought him the title of ''the 
Newton of France." With wonderful mastery of analysiSi 
Laplace attacked the pending problems in the application 
of the law of gravitation to celestial motions. During the 
succeeding fifteen years appeared most of his original contri- 
butions to astronomy. His career was one of almost uninter- 
rupted prosperity. In 1784 he succeeded B^zout as examiner 
to the royal artillery, and the following year he became mem- 
ber of the Academy of Sciences. He was made president of 
the Bureau of Longitude ; he aided in the introduction of the 
decimal system, and taught, with Lagrange, mathematics in 
the Ecole NormcUe, When, during the Bevolution, there arose 
a cry for the reform of everything, even of the calendar, 
Laplace suggested the adoption of an era beginning with the 
year 1250, when, according to his calculation, the major axis 
of the earth's orbit had been perpendicular to the equinoctial 
line. The year was to begin with the vernal equinox, and the 
zero meridian was to be located east of Paris by 185.30 degrees 
of the centesimal division of the quadrant, for by this meridian 
the beginning of his proposed era fell at midnight. But the 
revolutionists rejected this scheme, and made the start of the 
new era coincide with the beginning of the glorious French 
Republic.** 

Laplace was justly admired throughout Europe as a most 
sagacious and profound scientist, but, unhappily for his repu- 
tation, he strove not only after greatness in science, but also 
after political honours. The political career of this eminent 
scientist was stained by servility and suppleness. After the 
18th of Brumaire, the day when Napoleon was made emperor, 
Laplace's ardour for republican principles suddenly gave way 
to a great devotion to the emperor. Napoleon rewarded this 
devotion by giving him the post of minister of the interiory 



272 A HISTOBT OF MATHEMATICS. 

but dismissed him after six months for incapacity. Said 
Napoleon, '< Laplace ne saisissait aucune question sous son 
veritable point de vue; il cherchait des subtilit^s partout, 
n'avait que des id^s problematiques, et portait enfin I'esprit 
des infiniment petits jusque dans I'administration." Desirous 
to retain his allegiance, Napoleon elevated him to the Senate 
and bestowed various other honours upon him. Nevertheless, 
he cheerfully gave his voice in 1814 to the dethronement of 
his patron and hastened to tender his services to the Bourbons, 
thereby earning the title of marquis. This pettiness of his 
character is seen in his writings. The first edition of the 
Systhae du monde was dedicated to the Council of Five Hun- 
dred. To the third volume of the M^canique C&este is prefixed 
a note that of all the truths contained in the book, that most 
precious to the author was the declaration he thus made of 
gratitude and devotion to the peace-maker of Europe. After 
this outburst of affection, we are surprised to find in the editions 
of the TMorie analytique des probabilit^Sy which appeared after 
the Restoration, that the original dedication to the emperor is 
suppressed. 

Though supple and servile in politics, it must be said that 
in religion and science Laplace never misrepresented or con- 
cealed his own convictions however distasteful they might be 
to others. In mathematics and astronomy his genius shines 
with a lustre excelled by few. Three great works did he give 
to the scientific world, — the M4canique Cileste, the Exposition 
du syst^me du mondey and the Th4orie analytique des probabtli' 
ties. Besides these he contributed important memoirs to the 
French Academy. 

We first pass in brief review his astronomical researches. 
In 1773 he brought out a paper in which he proved that the 
mean motions or mean distances of planets are invariable or 
merely subject to small periodic changes. This was the first 



EULEB, LAGRANGE, AND LAPLACE. 273 

and most important step in establishing the stability of the 
solar system.** To Newton and also to Euler it had seemed 
doubtful whether forces so numerous, so variable in position, 
so different in intensity, as those in the solar system, could be 
capable of maintaining permanently a condition of equilibrium. 
Newton was of the opinion that a powerful hand must inter- 
vene from time to time to repair the derangements occa- 
sioned by the mutual action of the different bodies. This 
paper was the beginning of a series of profound researches by 
Lagrange and Laplace on the limits of variation of the various 
elements of planetary orbits, in which the two great mathema- 
ticians alternately surpassed and supplemented each other. 
Laplace's first paper really grew out of researches on the 
theory of Jupiter and Saturn. The behaviour of these planets 
had been studied by Euler and Lagrange without receiving 
satisfactory explanation. Observation revealed the existence 
of a steady acceleration of the mean motions of our moon and 
of Jupiter and an equally strange diminution of the mean mo- 
tion of Saturn. It looked as though Saturn might eventually 
leave the planetary system, while Jupiter would fall into the 
sun, and the moon upon the earth. Laplace finally succeeded 
in showing, in a paper of 1784-1786, that these variations 
(called the "great inequality") belonged to the class of ordi- 
nary periodic perturbations, depending upon the law of attrac- 
tion. The cause of so influential a perturbation was found in 
the commensurability of the mean motion of the two planets. 

In the study of the Jovian system, Laplace was enabled to 
determine the masses of the moons. He also discovered cer- 
tain very remarkable, simple relations between the movements 
of those bodies, known as " Laws of Laplace." His theory of 
these bodies was completed in papers of 1788 and 1789. 
These, as well as the other papers here mentioned, were pub- 
lished in the Memoirs pr^erUis par divers savans. The year 



274 A HISTORY OF HATHE&IATIOS. 

1787 was made memorable by Laplace's announcement that 
the lunar acceleration depended upon the secular changes in 
the eccentricity of the earth's orbit. This removed all doubt 
then existing as to the stability of the solar system. The uni- 
versal validity of the law of gravitation to explain all motion 
in the solar system was established. That system, as then 
known, was at last found to be a complete machine. 

In 1796 Laplace published his Exposition du systkme du 
monde, a non-mathematical popular treatise on astronomy, 
ending with a sketch of the history of the science. Li this 
work he enunciates for the first time his celebrated nebular 
hypothesis. A similar theory had been previously proposed 
by Kant in 1755, and by Swedenborg ; but Laplace does not 
appear to have been aware of this. 

Laplace conceived the idea of writing a work which should 
contain a complete analytical solution of the mechanical prob- 
lem presented by the solar system, without deriving from 
observation any but indispensable data. The result was the 
Micanique Celeste, which is a systematic presentation embrac- 
ing all the discoveries of Newton, Clairaut, D'Alembert, Euler, 
Lagp:ange, and of Laplace himself, on celestial mechanics. 
The first and second volumes of this work were published in 
1799 ; the third appeared in 1802, the fourth in 1805. Of the 
fifth volume. Books XI. and XII. were published in 1823; 
Books XIII., XIV., XV. in 1824, and Book XVI. in 1825. The 
first two volumes contain the general theory of the motions 
and figure of celestial bodies. The third and fourth volumes 
give special theories of celestial motions, — treating particu- 
larly of motions of comets, of our moon, and of other satel- 
lites. The fifth volume opens with a brief history of celestial 
mechanics, and then gives in appendices the results of the 
author's later researches. The M^canique Celeste was such a 
master-piece, and so complete, that Laplace's successors have 



EULEB, LAOBAKOE, AND LAPLACE. 276 

been able to add comparatively little. The general part of 
the work was translated into German by Joh. Karl Burk- 
hardt, and appeared in Berlin, 1800-1802. Nathaniel Bowditch 
brought out an edition in English, with an extensive com- 
mentary, in Boston, 1829-1839. The Micanique C&este is not 
easy reading. The difficulties lie, as a rule, not so much in the 
subject itself as in the want of verbal explanation. A compli- 
cated chain of reasoning receives often no explanation what- 
ever. Biot, who assisted Laplace in revising the work for 
the press, tells that he once asked Laplace some explanation 
of a passage in the book which had been written not long 
before, and that Laplace spent an hour endeavouring to recover 
the reasoning which had been carelessly suppressed with the 
remark, "II est facile de voir." Notwithstanding the impor- 
tant researches in the work, which are due to Laplace himself, 
it naturally contains a great deal that is drawn from his pred- 
ecessors. It is, in fact, the organised result of a century of 
patient toil. But Laplace frequently neglects to properly 
acknowledge the source from which he draws, and lets the 
reader infer that theorems and formulae due to a predecessor 
are really his own. 

We are told that when Laplace presented Napoleon with a 
copy of the Micanique Cilestey the latter made the remark, 
"M. Laplace, they tell me you have written this large book on 
the system of the universe, and have never even mentioned 
its Creator." Laplace is said to have replied bluntly, "Je 
n'avais pas besoin de cette hypoth^se-la." This assertion, 
taken literally, is impious, but may it not have been intended 
to convey a meaning somewhat different from its literal one ? 
Newton was not able to explain by his law of gravitation all 
questions arising in the mechanics of the heavens. Thus, 
being unable to show that the solar system was stable, and 
suspecting in fact that it was unstable, Newton expressed the 



276 A HISTOBY OF MATHEMATICS. 

opinion that the special intervention, from time to time, of a 
powerful hand was necessary to preserve order. Now Laplace 
was able to prove by the law of gravitation that the solar 
system is stable, and in that sense may be said to have felt 
no necessity for reference to the Almighty. 

We now proceed to researches which belong more properly 
to pure mathematics. Of these the most conspicuous are on 
the theory of probability. Laplace has done more towards 
advancing this subject than any one other investigator. He 
published a series of papers, the main results of which were 
collected in his Th4orie analytique des prohahiliUs, 1812. The 
third edition (1820) consists of an introduction and two books. 
The introduction was published separately under the title, 
Essai philosophique sur les proba^ilitis, and is an admirable 
and masterly exposition without the aid of analytical formulae 
of the principles and applications of the science. The first 
book contains the theory of generating functions, which are 
applied, in the second book, to the theory of probability. 
Laplace gives in his work on probability his method of 
approximation to the values of definite integrals. The solu- 
tion of linear differential equations was reduced by him to 
definite integrals. One of the most important parts of the 
work is the application of probability to the method of least 
squares, which is shown to give the most probable as well as 
the most convenient results. 

The first printed statement of the principle of least squares 
was made in 1806 by Legendre, without demonstration. Gauss 
had used it still earlier, but did not publish it until 1809. 
The first deduction of the law of probability of error that 
appeared in print was given in 1808 by Robert Adrain in the 
Analyst, a journal published by himself in Philadelphia.* 
Proofs of this law have since been given by Gauss, Ivory, 
Herschel, Hagen, and others ; but all proofs contain some 



BTJLBB, LAOBANGE, AND LAPLACE. 277 

point of difficulty. Laplace's proof is perhaps the most satis- 
factory. 

Laplace's work on probability is very difficult reading, par- 
ticularly the part on the method of least squares. The 
analytical processes are by no means clearly established or 
free from error. "No one was more sure of giving the 
result of analytical processes correctly, and no one ever took 
so little care to point out the various small considerations on 
which correctness depends" (De Morgan). 

Of Laplace's papers on the attraction of ellipsoids, the most 
important is the one published in 1785, and to a great extent 
reprinted in the third volume of the Micanique C^este. It 
gives an exhaustive treatment of the general problem of 
attraction of any ellipsoid upon a particle situated outside 
or upon its surface. Spherical harmonics, or the so-called 
''Laplace's coefficients," constitute a powerful analytic engine 
in the theory of attraction, in electricity, and magnetism. 
The theory of spherical harmonics for two dimensions had 
been previously given by Legendre. Laplace failed to make 
due acknowledgment of this, and there existed, in con- 
sequence, between the two great men, "a feeling more 
than coldness." The potential function, F, is much used by 
Laplace, and is shown by him to satisfy the partial differential 

equation ^:t + ^-^ + ^i? = ^- "^^^ is known as Laplace's 
dor d}f dsr 

equation, and was first given by him in the more complicated 
form whicli it assumes in polar co-ordinates. The notion 
of potential was, however, not introduced into analysis by 
Laplace. The honour of that achievement belongs to La- 
grange.* 

Among the minor discoveries of Laplace are his method of 
solving equations of the second, third, and fourth degrees, 
his memoir on singular solutions of differential equations, his 



278 A HISTORY OF MATEUBMATIOS. 

researches in finite differences and in determinants, the estab- 
lishment of the expansion theorem in determinants which had 
been previously given by Yandermonde for a special case, the 
determination of the complete integral of the linear differen- 
tial equation of the second order. In the Micanique CHeate he 
made a generalisation of Lagrange's theorem on the develop- 
ment of functions in series known as Laplace's theorem. 

Laplace's investigations in physics were quite extensive. 
We mention here his correction of Newton's formula on the 
velocity of sound in gases by taking into account the* changes 
of elasticity due to the heat of compression and cold of rarefac- 
tion ; his researches on the theory of tides ; his mathematical 
theory of capillarity ; his explanation of astronomical refrac- 
tion ; his formulae for measuring heights by the barometer. 

Laplace's writings stand out in bold contrast to those of 
Lagrange in their lack of elegance and symmetry. Laplace 
looked upon mathematics as the tool for the solution of physi- 
cal problems. The true result being once reached, he spent 
little time in explaining the various steps of his analysis, or 
in polishing his work. The last years of his life were spent 
mostly at Arcueil in peaceful retirement on a country-place, 
where he pursued his studies with his usual vigour until his 
death. He was a great admirer of Euler, and would often 
say, " Lisez Euler, lisez Euler, c'est notre maitre k tons." 

Abnit-Theophile Vandermonde (1735-1796) studied music 
during his youth in Paris and advocated the theory that all 
art rested upon one general law, through which any one could 
become a composer with the aid of mathematics. He was the 
first to give a connected and logical exposition of the theory 
of determinants, and may, therefore, almost be regarded as 
the founder of that theory. He and Lagrange originated the 
method of combinations in solving equations.* 

Adrien Marie Legendre (1752-1833) was educated at the 



BULEB, LAQBAN6E, AND LAPLACB. 279 

College Mazarin in Paris, where he began the study of mathe- 
matics under Abb^ Marie. His mathematical genius secured 
for him the position of professor of mathematics at the mili- 
tary school of Paris. While there he prepared an essay on 
the curve described by projectiles thrown into resisting media 
(ballistic curve), which captured a prize offered by the Royal 
Academy of Berlin. In 1780 he resigned his position in order 
to reserve more time for the study of higher mathematics. 
He was then made member of several public commissions. 
In 1795 he was elected professor at the Normal School and 
later was appointed to some minor government positions. 
Owing to his timidity and to Laplace's unfriendliness toward 
him, but few important public offices commensurate with his 
ability were tendered to him. 

As an analyst, second only to Laplace and Lagrange, Legen- 
dre enriched mathematics by important contributions, mainly 
on elliptic integrals, theory of numbers, attraction of ellip- 
soids, and least squares. The most important of Legendre's 
works is his Fonctiana eUiptiques, issued in two volumes in 
1825 and 1826. He took up the subject where Euler, Landen, 
and Lagrange had left it, and for forty years was the only one 
to cultivate this new branch of analysis, until at last Jacobi and 
Abel stepped in with admirable new discoveries." Legendre 
imparted to the subject that connection and arrangement 
which belongs to an independent science. Starting with an 
integral depending upon the square root of a polynomial of 
the fourth degree in x, he showed that such integrals can be 
brought back to three canonical forms, designated by F(^), 
E(<l>)f and II (<^), the radical being expressed in the form 
A(<^) = Vl — A:* sin'^. He also undertook the prodigious task 
of calculating tables of arcs of the ellipse for different degrees 
of amplitude and eccentricity, which supply the means of 
integrating a large number of differentials. 



280 A HISTOBY OF MATHEBiATICS. 

An earlier publication which contained part of his researches 
on elliptic functions was his CcUcul intigrcU in three volumes 
(1811, 1816, 1817), in which he treats also at length of the 
two classes of definite integrals named by him Etderian, He 
tabulated the values of log T(p) for values of p between 
land 2. 

One of the earliest subjects of research was the attraction 
of spheroids, which suggested to Legendre the function P^ 
named after him. His memoir was presented to the Academy 
of Sciences in 1783. The researches of Maclaurin and Lagrange 
suppose the point attracted by a spheroid to be at the surface 
or within the spheroid, but Legendre showed that in order to 
determine the attraction of a spheroid on any external point 
it suffices to cause the surface of another spheroid described 
upon the same foci to pass through that point Other memoirs 
on ellipsoids appeared later. 

The two household gods to which Legendre sacrificed with 
ever-renewed pleasure in the silence of his closet were the 
elliptic functions and the theory of numbers. His researches 
on the latter subject, together with the numerous scattered 
fragments on the theory of numbers due to his predecessors 
in this line, were arranged as far as possible into a systematic 
whole, and published in two large quarto volumes, entitled 
Thione des nombres, 1830. Before the publication of this 
work Legendre had issued at divers times preliminary articles. 
Its crowning pinnacle is the theorem of quadratic reciprocity, 
previously indistinctly given by Euler without proof, but for the 
first time clearly enunciated and partly proved by Legendre.** 

While acting as one of the commissioners to connect Green- 
wich and Paris geodetically, Legendre calculated all the tri- 
angles in France. This furnished the occasion of establishing 
formulae and theorems on geodesies, on the treatment of the 
spherical triangle as if it were a plane triangle, by applying 



EULGii, LAGBANGE, ASD LAPLACE, 



281 



[ certain coirections to the angles, and on the metbod of least 
' squares, published for the first time by him without demon- 
stration in 1806. 

Legendre wrote an Elimenta de Qiomitrie, 1794, which 
enjoyed great popularity, being generally adopted ou the 
Continent and in the United States as a substitute for Emilid. 
This great modern rival of Euclid passed through numerous 
editions ; the later ones containing the elements of trigonom- 
etry and a proof of the irrationality of x and »*. Much 
attention was given by Legendre to the subject of parallel 
lines. In the earlier editions of the Elhnetits, he made direct 
appeal to the senses for the correctness of tlie " parallel-aiioin." 
He then attempted to demonstrate that "axiom," but his 
proofs did not satisfy even himself. In VoL XIL of the 
Memoirs of the Institute is a paper by Legendre, containing 
I his last attempt at a solution of the problem. Assoming 
I apace to be infinite, he proved satisfactorily that it is impossible 
I for the sum of the three angles of a triangle to exceed two 
I right angles ; and that if there be any triangle the sum of 
« angles is t ' - ~ <'ien the same must be trrui 

1 triuqClBS. ^tcp, to show that this Bum 

ii;li«, Ilia demonstration neces- 
Q of the thn-« 
the theory of 



282 A HISTORY OF MATHEMATICS. 

being n oble^. c.mil(l not ftnter thftjar tillery. althoug h he were 
a second Newton."" He was soon appointed totne matbe- 

' matical chairTn the military school. At the age of twenty- 
one he went to Paris to read before the Academy of Sciences 
a memoir on the resolution of numerical equations , which 
^ 2J was an improvement on N ewWs method of jp proximation . 
This investigation of his early youtli he never lost sight of. 
He lectured upon it in the Polytechnic School ; he developed 
it on the banks of the Kile ; it constituted a part of a work 
entitled Analyse des equaJtionea determines (1831), which was 
in press when death overtook him. This work contained 
" Four ier's theorem" on the numhfir. o|_realroote between 
twochosen limits. Budan had published this result as early 
as 1807, but thereTs evidence to show that Fourier had estab- 
lished it before Budan's publication. These brilliant result s 
we re ec lipsed by the theoiem jof Stur m, published in 1835 . 

Fourier took a prominent part at his home in promoting 
the Revolution. Under the French Revolution the arts and 
sciences seemed for a time to flourish. The reformation of 
the weights and measures was planned with grandeur of con- 
ception. The Normal School was created in 1795, of which 
Fourier became at first pupil, then lecturer. His brilliant 
success secured him a chair in the Polytechnic School, the 
duties of which he afterwards quitted, along with Monge and 
Berthollet, to accompany Napoleon on his campaign to Egypt. 
Napoleon founded the Institute of Egypt, of which Fourier 
became secretary. In Egypt he engaged not only in scientific 
work, but discharged important political functions. After 
his return to France he held for fourteen years the prefecture 
of Grenoble. During this period he carried on his elaborate 

_^ investigations on the propagation of heat in solid bodies, 

published in 1822 in his work entitled La Theorie Analyiique 

~ dela CTuUeur. This work marks an epoch in the history of 



BULBB, LAGBAKGE, AND LAPLACB. 288 

mathematical physics. '' Fourier's series'' constitates its 
gem. By this research a long controversy was brought to a 
close, and the fact established that aiiy arbitrary function 
can be represented by a trigonometric series. The first 
announcement of this great discovery was made by Fourier 
in 1807, before the French Academy. The trigonometric 

series 2 (a^ sin nx + &« cos na;) represents the function ^ (a;) 

for every value of x, if the coefficients a.= - I ^(x) sianxdx, 

and h^ be equal to a similar integral. The weak point in 
Fourier's analysis lies in his failure to prove generally that 
the trigonometric series actually converges to the value of 
^e function. In 1827 Fourier succeeded Laplace as president 
of the council of the Polytechnic School. 

Before proceeding to the origin of modern geometry we shall 
speak briefly of the introduction of higher analysis into Great 
Britain. This took place during the first quarter of this cen- 
tury. The British began to deplore the very small progress 
that science was making in England as compared with its 
racing progress on the Continent. In 1813 the " Analytical \y 
Society" was formed at Cambridge. This was a smaHcliib ~ 
established by George Peacock, John Herschel, Charles Bab- 
bage, and a few other Cambridge students, to promote, as it 
was humorously expressed, the principles of pure "D-ism," 
that is, the Leibniziau notation in the calculus against those 
of "dot-age," or of the Newtonian notation. This struggle 
ended in the introduction into Cambridge of the notation 

-^, to the exclusion of the fluxional notation y. This 
dx 

was a great step in advance, not on account of any great 
sux>eriority of the Leibnizian over the Newtonian notation, 
but because the adoption of the former opened up to English 
students the vast storehouses of continental discoveries.. Sir 
William Thomson, Tait, and some other modern writers find 



/ 



284 A HISTORY OF MATHEMATICS. 

it frequently convenient to use both notations. Herschel, 
Peacock, and Babbage translated, in 1816, from the French, 
Lacroix's treatise on the differential and integral calculus, and 
added in 1820 two volumes of examples. Lacroix's was one 
of the best and most extensive works on the calculus of that 
time. Of the three founders of the "Analytical Society," 
Peacock afterwards did most work in pure mathematics. 
Babbage J^came famous for his invention of a cal culating 
engine superior to Pascal's. It was never finished, owine: 
to a misunderstanding with the government, and a conse- 
quent failure to secure funds. John Herschel, the eminent 
astronomer, displayed his mastery over higher analysis in 
memoirs communicated to the Royal Society on new applica- 
tions of mathematical analysis, and in articles contributed 
to cyclopaedias on light, on meteorology, and on the history 
of mathematics. 

George Peacock (1791-1858) was educated at Trinity College, 
Cambridge, became Lowndean professor there, and later, dean 
of Ely. His chief publications are his Algebra, 1830 and 1842, 
and his Report on Recent Progress in Analysis, which was the 
first of several valuable summaries of scientific progress printed 
in the volumes of the British Association. He was one of the 
first to study seriously the fundamental principles of algebra, 
and to fully recognise its purely symbolic character. He 
advances, though somewhat imperfectly, the " principle of the 
permanence of equivalent forms." It assumes that the rules 
applying to the symbols of arithmetical algebra apply also 
in symbolical algebra. About this time D. F. Gregory wrote 
a paper "on the real nature of symbolical algebra," which 
brought out clearly the commutative and distributive laws. 
These laws had been noticed years before by the inventors 
of symbolic methods in the calculus. It was Servois who 
introduced the names commtUative and distributive in 1813. 



EULEB, LAGBANGE, AND LAPLACE. 285 

Peacock's investigations on the foundation of algebra were 
considerably advanced by Be Morgan and Hankel. 

James Ivory (1765-1845) was a Scotch mathematician who 
for twelve years, beginning in 1804, held the mathematical 
chair in the !Royal Military College at Marlow (now at Sand- 
hurst). He was essentially a self-trained mathematician, and 
almost the only one in Great Britain previous to the organisa- 
tion of the Analytical Society who was well versed in conti- 
nental mathematics. Of importance is his memoir (PhU 
Trans., 1809) in which the problem of the attraction of a 
homogeneous ellipsoid upon an external point is reduced to 
the simpler problem of the attraction of a related ellipsoid 
upon a corresponding point interior to it. This is known as 
" Ivory's theorem." He criticised with undue severity Laplace's 
solution of the method of least squares, and gave three proofs 
of the principle without recourse to probability ; but they are 
far from being satisfactory. 

The Origin of Modem Chometry. 

By the researches of Descartes and the invention of the cal- 
culus, the analytical treatment of geometry was brought into 
great prominence for over a century. Notwithstanding the 
efforts to revive synthetic methods made by Desargues, Pas- 
cal, De Lahire, Newton, and Maclaurin, the analytical method 
retained almost undisputed supremacy. It was reserved for 
the genius of Monge to bring synthetic geometry in the 
foreground, and to open up new avenues of progress. His 
O^om^trie descriptive marks the beginning of a wonderful 
development of modem geometry. 

Of the two leading problems of descriptive geometry, the 
one — to represent by drawings geometrical magnitudes — was 
brought to a high degree of perfection before the time of 



286 A HISTORY OF MATHEMATICS. 

Monge; the other — to solve problems on figures in space 
by constructions in a plane — had received considerable at- 
tention before his time. His most noteworthy predecessor 
in descriptive geometry was the Frenchman Fr&ier (1682- 
1773). But it remained for Monge to create descriptive 
geometry as a distinct branch of science by imparting to it 
geometric generality and elegance. All problems previously 
treated in a special and uncertain manner were referred 
back to a few general principles. He introduced the line 
of intersection of the horizontal and the vertical plane as 
the axis of projection. By revolving one plane into the 
other around this axis or ground-line, many advantages were 
gained." 

Gaspard Monge (1746-1818) was born at Beaune. The con- 
struction of a plan of his native town brought the boy imder 
the notice of a colonel of engineers, who procured for him an 
appointment in the college of engineers at M^zi^res. Being 
of low birth, he could not receive a commission in the army, 
but he was permitted to enter the annex of the school, where 
surveying and drawing were taught. Observing that all the 
operations connected with the construction of plans of fortifi- 
cation were conducted by long arithmetical processes, he sub- 
stituted a geometrical method, which the commandant at first 
refused even to look at, so short was the time in which it 
could be practised ; when once examined, it was received with 
avidity. Monge developed these methods further and thus 
created his descriptive geometry. Owing to the rivalry 
between the French military schools of that time, he was not 
permitted to divulge his new methods to any one outside of 
this institution. In 1768 he was made professor of mathemat- 
ics at M^zi^res. In 1780, when conversing with two of his 
pupils, S. F. Lacroix and Gayvemon in Paris, he was obliged 
to say, " All that I have here done by calculation, I could have 



BULBB, LAQBAN6E, AND LAPLAOB. 287 

done with the ruler and compass, but I am not allowed to 
reveal these secrets to you." But Lacroix set himself to 
examine what the secret could be, discovered the processes, and 
published them in 1795. The method was published by Monge 
himself in the same year, first in the form in which the short- 
hand writers took down his lessons given at the Normal School, 
where he had been elected professor, and then again, in revised 
form, in the Journal des 4coles normcUes. The next edition 
occurred in 1798-1799. After an ephemeral existence of only 
four months the Normal School was closed in 1795. In the 
same year the Polytechnic School was opened, in the estab- 
lishing of which Monge took active part. He taught there 
descriptive geometry until his departure from France to accom- 
pany Napoleon on the Egyptian campaign. He was the first 
president of the Institute of Egypt. Monge was a zealous 
partisan of Napoleon and was, for that reason, deprived of all 
his honours by Louis XVIII. This and the destruction of the 
Polytechnic School preyed heavily upon his mind. He did 
not long survive this insult. 

Mongers numerous papers were by no means confined to de- 
scriptive geometry. His analytical discoveries are hardly less 
remarkable. He introduced into analytic geometry the me- 
thodic use of the equation of a line. He made important 
contributions to surfaces of the second degree (previously 
studied by Wren and Euler) and discovered between the 
theory of surfaces and the integration of partial differential 
equations, a hidden relation which threw new light upon both 
subjects. He gave the differential of curves of curvature, 
established a general theory of curvature, and applied it to the 
ellipsoid. He found that the validity of solutions was not 
impaired when imaginaries are involved among subsidiary 
quantities. Monge published the following books: SicUicM, 
1786 ; Apjdications de Valg^e d la gionUtrie, 1805 ; Apptieor 



288 A HISTOBY OF MATHEMATICS. 

tion de Vanalyse d la gSom^rie, The last two contain most of 
his miscellaneous papers. 

Monge was an inspiring teacher, and he gathered around 
him a large circle of pupils, among which were Dupin, Servois, 
Brianchion, Hachette, Biot, and Poncelet. 

Charles Dupin (1784-1873), for many years professor of 
mechanics in the Conservatoire des Arts et Metiers in Paris, 
published in 1813 an important work on D&vdoppements de 
g4omitrie, in which is introduced the conception of conjugate 
tangents of a point of a surface, and of the indicatrix." It 
contains also the theorem known as "Dupin's theorem." 
Surfaces of the second degree and descriptive geometry were 
successfully studied by Jean Nicolas Pierre Hachette (1769- 
1834), who became professor of descriptive geometry at the 
Polytechnic School after the departure of Monge for Rome and 
Egypt. In 1822 he published his Traits de gioin^trie descriptive. 

Descriptive geometry, which arose, as we have seen, in 
technical schools in France, was transferred to Grermany at 
the foundation of technical schools there. G. Schreiber, 
professor in Karlsruhe, was the first to spread Monge's 
geometry in Germany by the publication of a work thereon 
in 1828-1829." In the United States descriptive geometry was 
introduced in 1816 at the Military Academy in West Point 
by Claude Crozet, once a pupil at the Polytechnic School in 
Paris. Crozet wrote the first English work on the subject.* 

Lazare Nicholas Marguerite Camot (1753-1823) was born at 
Nolay in Burgundy, and educated in his native province. 
He entered the army, but continued his mathematical studies, 
and wrote in 1784 a work on machines, containing the earliest 
proof that kinetic energy is lost in collisions of bodies. With 
the advent of the Revolution he threw himself into politics, 
and when coalesced Europe, in 1793, launched against France 
a million soldiers, the gigantic task of organising fourteen 



EULEB, LAGRANGE, AND LAPLACE. 289 

armies to meet the enemy was achieved by him. He was 
banished in 1796 for opposing Napoleon's coup d'itat. The 
refugee went to Geneva, where he issued, in 1797, a work 
still frequently quoted, entitled, B4flexion8 aur la M^aphysique 
du Calcul Infinitisimal. He declared himself as an " irrecon- 
cilable enemy of kings." After the Eussian campaign he 
offered to fight for France, though not for the empire. On 
the restoration he was exiled. He died in Magdeburg. His 
Oiomitrie de position, 1803, and his Essay on Transversals^ 
1806, are important contributions to modem geometry. While 
Monge revelled mainly in three-dimensional geometry, Garnot 
confined himself to that of two. By his effort to explain 
the meaning of the negative sign in geometry he established 
a "geometry of position," which, however, is different from 
the "Geometric der Lage" of to-day. He invented a class 
of general theorems on projective properties of figures, which 
have since been pushed to great extent by Poncelet, Chasles, 
and others. 

Jean Victor Poncelet (1788-1867), a native of Metz, took 
part in the Russian campaign, was abandoned as dead on the 
bloody field of Krasnoi, and taken prisoner to Saratoff. De- 
prived there of all books, and reduced to the remembrance 
of what he had learned at the Lyceum at Metz and the Poly- 
technic School, where he had studied with predilection the 
works of Monge, Garnot, and Brianchion, he began to study 
mathematics from its elements. He entered upon original 
researches which afterwards made him illustrious. While 
in prison he did for mathematics what Bunyan did for 
literature, — produced a much-read work, which has remained 
of great value down to the present time. He returned to 
France in 1814, and in 1822 published the work in question, 
entitled, Traits des Propriitis projectives des figures. In it 
he investigated the properties of figures which remain un- 



290 A mSTOBY OF MATHEMATICS. 

altered by projection of the figures. The projection is not 
effected here by parallel rays of prescribed direction, as with 
Monge, but by central projection. Thus perspective projec- 
tion, used before him by Desargues, Pascal, Kewton, and Lam- 
bert, was elevated by him into a fruitful geometric method. 
In the same way he elaborated some ideas of De Lahire, 
Servois, and Gergonne into a regular method — the method 
of " reciprocal polars." To him we owe the Law of Duality 
as a consequence of reciprocal polars. As an independent 
principle it is due to Grergonne. Poncelet wrote much on 
applied mechanics. In 1838 the Faculty of Sciences was 
enlarged by his election to the chair of mechanics. 

While in France the school of Monge was creating modem 
geometry, efforts were made in England to revive Greek 
geometry by Robert Simson (1687-1768) and Biatthew Stewart 
(1717-1785). Stewart was a pupil of Simson and Maclaurin, 
and succeeded the latter in the chair at Edinburgh. During 
the eighteenth century he and Maclaurin were the only promi- 
nent mathematicians in Great Britain. His genius was ill- 
directed by the fashion then prevalent in England to ignore 
higher analysis. In his Four Tracts, Physical and Mathe- 
matical, 1761, he applied geometry to the solution of difficult 
astronomical problems, which on the Continent were ap- 
proached analytically with greater success. He published, in 
1746, General Theorems, and in 1763, his Propositiones geo- 
metrical more veterum demonstrator. The former work con- 
tains sixty-nine theorems, of which only five are accompanied 
by demonstrations. It gives many interesting new results 
on the circle and the straight line. Stewart extended some 
theorems on transversals due to Giovanni Ceva (1648-1737), 
an Italian, who published in 1678 at Mediolani a work con- 
taining the theorem now known by his name. 



RECENT TIMES. 



Keyer more zealously and successfally has mathematics 
been cultivated than in this century. Nor has progress, as 
in previous periods, been confined to one or two countries. 
While the French and Swiss, who alone during the preceding 
epoch carried the torch of progress, have continued to develop 
mathematics with great success, from other countries whole 
armies of enthusiastic workers have wheeled into the front 
rank. Germany awoke from her lethargy by bringing for- 
ward Gauss, Jacobi, Dirichlet, and hosts of more recent men ; 
Great Britain produced her De Morgan, Boole, Hamilton, 
besides champions who are still living; Eussia entered the 
arena with her Lobatchewsky ; Norway with Abel; Italy with 
Cremona ; Hungary with her two Bolyais ; the United States 
with Benjamin Peirce. 

The productiveness of modem writers has been enormous. 
" It is diflScult," says Professor Cayley,** " to give an idea of 
the vast extent of modem mathematics. This word ' extent ' 
is not the right one : I mean extent crowded with beautiful 
detail, — not an extent of mere uniformity such as an object- 
less plain, but of a tract of beautiful country seen at first in 
the distance, but which will bear to be rambled through and 
studied in every detail of hillside and valley, stream, rock, 
wood, and flower." It is pleasant to the mathematician to 

think that in his, as in no other science, the achievements of 

291 



292 A HISTORY OF MATHEMATICS. 

every age remain possessions forever ; new discoveries seldom 
disprove older tenets ; seldom is anything lost or wasted. 

If it be asked wherein the utility of some modern exten- 
sions of mathematics lies, it must be acknowledged that it is 
at present difficult to see how they are ever to become appli- 
cable to questions of common life or physical science. But 
our inability to do this should not be urged as an argument 
against the pursuit of such studies. In the first place, we 
know neither the day nor the hour when these abstract 
developments will find application in the mechanic arts, in 
physical science, or in other branches of mathematics. For 
example, the whole subject of graphical statics, so useful 
to the practical engineer, was made to rest upon von Staudt's 
Oeometrie der Lage; Hamilton's " principle of varying action " 
has its use in astronomy; complex quantities, general inte- 
grals, and general theorems in integration offer advantages in 
the study of electricity and magnetism. " The utility of such 
researches," says Spottiswoode,*^ "can in no case be discounted, 
or even imagined beforehand. Who, for instance, would have 
supposed that the calculus of forms or the theory of substitu- 
tions would have thrown much light upon ordinary equations ; 
or that Abelian functions and hyperelliptic transcendents 
would have told us anything about the properties of curves ; 
or that the calculus of operations would have helped us in 
any way towards the figure of the earth ? " A second reason 
in favour of the pursuit of advanced mathematics, even when 
there is no promise of practical application, is this, that math- 
ematics, like poetry and music, deserves cultivation for its 
own sake. 

The great characteristic of modern mathematics is its gen- 
eralising tendency. Nowadays little weight is given to iso- 
lated theorems, " except as affording hints of an unsuspected 
new sphere of thought, like meteorites detached from some 



SYNTHETIC 6B0METBY. 298 

undiscovered planetary orb of speculation." In mathematics, 
as in all true sciences, no subject is considered in itself alone, 
but always as related to, or an outgrowth of, other things. 
The development of the notion of continuity plays a leading 
part in modem research. In geometry the principle of con- 
tinuity, the idea of correspondence, and the theory of projec- 
tion constitute the fundamental modem notions. Continuity 
asserts itself in a most striking way in relation to the circular 
points at infinity in a plane. In algebra the modem idea finds 
expression in the theory of linear transformations and invari- 
ants, and in the recognition of the value of homogeneity and 
symmetry. 



SYNTHETIC GEOMETRY. 

The conflict between geometry and analysis which arose 
near the close of the last century and the beginning of the 
present has now come to an end. Neither side has come 
out victorious. The greatest strength is found to lie, not in 
the suppression of either, but in the friendly rivalry between 
the two, and in the stimulating influence of the one upon the 
other. Lagrange prided himself that in his Mecanique Anor 
lytique he had succeeded in avoiding all figures ; but since his 
time mechanics has received much help from geometry. 

Modem synthetic geometry was created by several investi- 
gators about the same time. It seemed to be the outgrowth 
of a desire for general methods which should serve as threads 
of Ariadne to guide the student through the labyrinth of theo- 
rems, corollaries, porisms, and problems. Synthetic geometry 
was first cultivated by Monge, Carnot, and Poncelet in France ; 
it then bore rich fruits at the hands of Mobius and Steiner in 
(jermany and Switzerland, and was finally developed to still 



294 A HISTORY OF MATHEMATICS. 

higher perfection by Chasles in France^ von Standi in (Ger- 
many, and Cremona in Italy. 

Augustus Ferdinand Mobius (1790-1868) was a native of 
Schulpforta in Prussia. He studied at G6ttingen under 
Gauss, also at Leipzig and Halle. In Leipzig he became, in 
1815, privat-docent, the next year extraordinary professor of 
astronomy, and in 1844 ordinary professor. This position he 
held till his death. The most important of his researches 
are on geometry. They appeared in Crdle's Journal^ and in 
his celebrated work entitled Der BarycerUriache CalctU, Leipzig, 
1827. As the name indicates, this calculus is based upon 
properties of the centre of gravity.* Thus, that the point 
S is the centre of gravity of weights a,b,Cfd placed at the 
points A, B, C, D respectively, is expressed by the equation 

(a + b + c + d)S = aA + bB + cO+dD. 

His calculus is the beginning of a quadruple algebra, and con- 
tains the germs of Grassmann's marvellous system. In desig- 
nating segments of lines we find throughout this work for the 
first time consistency in the distinction of positive and nega- 
tive by the order of letters AB, BA, Similarly for triangles 
and tetrahedra. The remark that it is always possible to give 
three points Ay By C such weights a, Py y that any fourth point 
M in their plane will become a centre of mass, led Mobius 
to a new system of co-ordinates in which the position of a 
point was indicated by an equation, and that of a line by 
co-ordinates. By this algorithm he found by algebra many 
geometric theorems expressing mainly invariantal properties, 
— for example, the theorems on the anharmonic relation. 
Mobius wrote also on statics and astronomy. He generalised 
spherical trigonometry by letting the sides or angles of tri- 
angles exceed 180°. 



SYNTHETIC 6E0MBT&Y. 295 

Jacob Steiner (1796-1863), 'Hhe greatest geometrician since 
the time of Euclid," was bom in Utzendorf in the Canton of 
Bern. He did not learn to write till he was fourteen. At 
eighteen he became a pupil of Pestalozzi. Later he studied 
at Heidelberg and Berlin. When Crelle started, in 1826, the 
celebrated mathematical journal bearing his name, Steiner and 
Abel became leading contributors. In 1832 Steiner published 
his Systematische Entwickelung der Abhdngigkeit geometrischer 
OestdUen von einander, ^^ in which is uncovered the organism 
by which the most diverse phenomena {Erscheinungen) in 
the world of space are united to each other." Through the 
influence of Jacobi and others, the chair of geometry was 
founded for him at Berlin in 1834. This position he occupied 
until his death, which occurred after years of bad health. In 
his Systematische Entwickelungeny for the first time, is the 
principle of duality introduced at the outset. This book and 
von Staudt's lay the foundation on which synthetic geometry 
in its present form rests. Not only did he fairly complete the 
theory of curves and surfaces of the second degree, but he 
made great advances in the theory of those of higher degrees. 
In his hands synthetic geometry made prodigious progress. 
New discoveries followed each other so rapidly that he often 
did not take time to record their demonstrations. In an 
article in Crelle' s Journal on Allgemeine Eigenschaften Alge- 
braischer Curven he gives without proof theorems which were 
declared by Hesse to be "like Fermat's theorems, riddles to 
the present and future generations." Analytical proofs of 
some of them have been given since by others, but Cremona 
finally proved them all by a synthetic method. Steiner dis- 
covered synthetically the two prominent properties of a sur- 
face of the third order; viz. that it contains twenty-seven 
straight lines and a pentahedron which has the double points 
for its vertices and the lines of the Hessian 6f the given sur- 



296 A HI8T0BY OF MATHEMATICS. 

face for its edges." The first property was discovered anar 
lytically somewhat earlier in England by Cayley and Salmon, 
and the second by Sylvester. Steiner^s work on this subject 
was the starting-point of important researches by H. Schroter, 
F. August, L. Cremona, and K. Sturm. Steiner made investi- 
gations by synthetic methods on maxima and minima, and 
arrived at the solution of problems which at that time alto- 
gether surpassed the analytic power of the calculus of varia- 
tions. He generalised the hexagrammum mysHcum and also 
Malfatti's problem.* Malf atti, in 1803, proposed the problem, 
to cut three cylindrical holes out of a three-sided priam in 
such a way that the cylinders and the prism have the same 
altitude and that the volume of the cylinders be a maximum. 
This problem was reduced to another, now generally known 
as Half atti's problem : to inscribe three circles in a triangle 
that each circle will be tangent to two sides of a triangle and 
to the other two circles. Malfatti gave an analytical solution, 
but Steiner gave without proof a construction, remarked that 
there were thirty-two solutions, generalised the problem by 
replacing the three lines by three circles, and solved the 
analogous problem for three dimensions. This general prob- 
lem was solved analytically by C. H. Schellbach (1809-1892) 
and Cayley, and by Clebsch with the aid of the addition 
theorem of elliptic functions." 

Steiner^s researches are confined to synthetic geometry. He 
hated analysis as thoroughly as Lagrange disliked geometry. 
Steiner^s Gesammelte Werke were published in Berlin in 1881 
and 1882. 

Michel Chasles (1793-1880) was born at Epernon, entered 
the Polytechnic School of Paris in 1812, engaged afterwards 
in business, which he later gave up that he might devote all 
his time to scientific pursuits. In 1841 he became professor of 
geodesy and mechanics at the Polytechnic School; later, 



SYNTHETIC GEOMETRY. 297 

** Professeur de Gr^om^trie sup^rieure k la Faculty des Sciences 
de Paris." He was a voluminous writer on geometrical sub- 
jects. In 1837 he published his admirable Aper^ historique 
8ur Vcmgine et le d^veloppement des m^thodes en g4om6trie, con- 
taining a history of geometry and^ as an appendix, a treatise 
"sur deux principes g^n^raux de la Science." The Aper^ 
historique is still a standard historical work; the appendix 
contains the general theory of Homography (Collineation) and 
of duality (Reciprocity). The name duality is due to Joseph 
Diaz (rergonne (1771-1859). Chasles introduced the term 
anharmanic ratio, corresponding to the German Doppelver- 
haUniss and to Clifford's cross-raJtio. Chasles and Steiner 
elaborated independently the modem synthetic or projective 
geometry. Numerous original memoirs of Chasles were pub- 
lished later in the Journal de VEcole Polytechnique, He gave 
a reduction of cubics, different from Newton's in this, that the 
five curves from which all others can be projected are sym- 
metrical with respect to a centre. In 1864 he began the pub- 
lication, in the Comptes rendu^, of articles in which he solves 
by his " method of characteristics " and the " principle of cor- 
respondence" an immense number of problems. He deter- 
mined, for instance, the number of intersections of two curves 
in a plane. The method of characteristics contains the basis 
of enumerative geometry. The application of the principle of 
correspondence was extended by Cayley, A. Brill, H. G. Zeu- 
then, H. A. Schwarz, G. H. Halphen (1844-1889), and others. 
The full value of these principles of Chasles was not brought 
out until the appearance, in 1879, of the Kalkul der Abzdhl- 
enden Geometrie by Hermann Schubert of Hamburg. This 
work contains a masterly discussion of the problem of enumer- 
ative geometry, viz. to determine how many geometric figures 
of given definition satisfy a sufficient number of conditions. 
Schubert extended his enumerative geometry to n-dimensional 
space." 



298 A UlSTOKY OF MATHEMATICS. 

To Chasles we owe tlie introduction into projective geometry 
of non-projective properties of figures by means of the infi- 
nitely distant imaginary sphero-circle.*^ Remarkable is his 
complete solution, in 1846, by synthetic geometry, of the 
difficult question of the attraction of an ellipsoid on an exter- 
nal point. This was accomplished analytically by Poisson in 
1835. The labours of Chasles and Steiner raised synthetic 
geometry to an honoured and respected position by the side 
of analysis. 

Karl Georg Christian von Staudt (1798-1867) was bom in 
Kothenburg on the Tauber, and, at his death, was professor 
in Erlangen. His great works are the Oeometrie der Lage^ 
Niirnberg, 1847, and his Beitrdge zur Oeometrie der Lage, 1856- 
1860. The author cut loose from algebraic formulae and from 
metrical relations, particularly the anharmonic ratio of Steiner 
and Chasles, and then created a geometry of position, which 
is a complete science in itself, independent of all measure- 
ments. He shows that projective properties of figures have 
no dependence whatever on measurements, and can be estab- 
lished without any mention of them. In his theory of what 
he calls "Wurfe," he even gives a geometrical definition of 
a number in its relation to geometry as determining the posi- 
tion of a point. The Beitrdge contains the first complete and 
general theory of imaginary points, lines, and planes in pro- 
jective geometry. Representation of an imaginary point is 
sought in the combination of an involution with a determi- 
nate direction, both on the real line through the point. 
While purely projective, von Staudt's method is intimately 
related to the problem of representing by actual points and 
lines the imaginaries of analytical geometry. This was sys- 
tematically undertaken by C. F. Maximilien Marie, who worked, 
however, on entirely different lines. An independent attempt 
has been made recently (1893) by F. H. Loud of Colorado 



SYNTHETIC GEOMBTRY. 299 

College. Von Staudt's geometry of position. was for a long 
time disregarded, mainly, no doubt, because his book is 
extremely condensed. An impulse to the study of this subject 
was given by Gulmann, who rests his graphical statics upon 
the work of von Staudt. An interpreter of von Staudt was 
at last found in Theodor Keye of Strassburg, who wrote a 
Oeometrie der Lage in 1868. 

Synthetic geometry has been studied with much success by 
Luigi Cremona, professor in the University of Kome. In 
his IrUroduzione ad una teoria geometrica delle curve piane 
he developed by a uniform method many new results and 
proved synthetically all important results reached before that 
time by analysis. Kis writings have been translated into 
(jerman by M. Curtze, professor at the gymnasium in Thorn. 
The theory of the transformation of curves and of the corre- 
spondence of points on curves was extended by him to three 
dimensions. Ruled surfaces, surfaces of the second order, 
space-curves of the third order, and the general theory of 
surfaces have received much attention at his hands. 

Karl Culmann, professor at the Polytechnicum in ZCirich^ 
published an epoch-making work on Die graphische Statik, 
Zdrich, 1864, which has rendered graphical statics a great 
rival of analytical statics. Before Culmann, B. E. Cousinery 
had turned his attention to the graphical calculus, but he 
made use of perspective, and not of modern geometry." Cul- 
mann is the first to undertake to present the graphical calculus 
as a symmetrical whole, holding the same relation to the new 
^geometry that analytical mechanics does to higher analysis. 
He makes use of the polar theory of reciprocal figures as 
expressing the relation between the force and the funicular 
polygons. He deduces this relation without leaving the plane 
of the two figures. But if the polygons be regarded as pro- 
jections of lines in space, these lines may be treated as recipro- 



300 A HISTORY OF MATHEMATICS. 

cal elements of a " Nullsystem." This was done by Clerk 
Maxwell in 1864, and elaborated further by Cremona.^ The 
graphical calculus has been applied by 0. Mohr of Dresden 
to the elastic line for continuous spans. Henry T. Eddy^ of 
the Rose Polytechnic Institute, gives graphical solutions of 
problems on the maximum stresses in bridges under concen- 
trated loads, with aid of what he calls "reaction polygons." 
A standard work. La Statique graphique, 1874, was issued by 
Maurice Levy of Paris. 

Descriptive geometry (reduced to a science by Monge in 
France, and elaborated further by his successors, Hachettey 
Dupin, OlivieTy J, de la Qoumerie) was soon studied also in 
other countries. The French directed their attention mainly 
to the theory of surfaces and their curvature; the (rermans 
and Swiss, through Schreiber, Pohlke, Schlessinger, and par- 
ticularly Fiedler, interwove projective and descriptive geome- 
try. Bellavitis in Italy worked along the same line. The 
theory of shades and shadows was first investigated by the 
French writers just quoted, and in Germany treated most 
exhaustively by Burmester." 

During the present century very remarkable generalisations 
have been made, which reach to the very root of two of the 
oldest branches of mathematics, — elementary algebra and 
geometry. In algebra the laws of operation have been ex- 
tended; in geometry the axioms have been searched to the 
bottom, and the conclusion has been reached that the space 
defined by Euclid's axioms is not the only possible non- 
contradictory space. Euclid proved (I. 27) that " if a straight, 
line falling on two other straight lines make the alternate 
angles equal to one another, the two straight lines shall be 
parallel to one another." Being unable to prove that in every 
other case the two lines are not parallel, he assumed this to 
be true in what is generally called the 12th " axiom," by some 



8TKTHBTIC 6E0METBT. 801 

the 11th << axiom." But this scHsalled axiom is far from 
axiomatic. After centuries of desperate but fruitless attempts 
to prove Euclid's assumption, the bold idea dawned upon 
the minds of several mathematicians that a geometry might 
be built up without assuming the parallel-axiouL While 
Legendre still endeavoured to establish the axiom by rigid 
proof, Lobatchewsky brought out a publication which assumed 
the contradictory of that axiom, and which was the first of 
a series of articles destined to clear up obscurities in the 
fundamental concepts, and to greatly extend the field of 
geometry. 

Kicholaus Ivanovitch Lobatchewsky (179S-1856) was bom at 
Makarief, in Nischni-Nowgorod, Russia, studied at Kasan, and 
from 1827 to 1846 was professor and rector of the University 
of Kasan. His views on the foundation of geometry were 
first made public in a discourse before the physical and mathe- 
matical faculty at Kasan, and first printed in the Kasan 
Messenger for 1829, and then in the Oelehrte Schriften der 
UniversMt Kasan, 1836-1838, under the title, " New Elements 
of Geometry, with a complete theory of Parallels." Being 
in the Russian language, the work remained unknown to 
foreigners, but even at home it attracted no notice. In 1840 
he published a brief statement of his researches in Berlin. 
Lobatchewsky constructed an ''imaginary geometry," as he 
called it, which has been described by Clifford as ''quite 
simple, merely Euclid without the vicious assumption." A 
remarkable part of this geometry is this, that through a 
point an indefinite number of lines can be drawn in a plane, 
none of which cut a given line in the same plane. A similar 
system of geometry was deduced independently by the Bolyais 
in Hungary, who called it " absolute geometry." 

Wolfgang Bolyai de Bolya (1775-1856) was bom in Szekler- 
Land, Transylvania. After studying at Jena, he went to 



802 A HISTORY OF MATHEMATICS. 

Gdttingen, where he became iDtimate with Gauss^ then nine- 
teen years old. Gauss used to say that Bolyai was the only 
man who fully understood his views on the metaphysics of 
mathematics. Bolyai became professor at the Reformed Col- 
lege of Maros-Y^^hely^ where for forty-seven years he had 
for his pupils most of the present professors of Transylvania. 
The first publications of this remarkable genius were dramas 
and poetry. Glad in old-time planter's garb, he was truly 
original in his private life as well as in his mode of thinking. 
He was extremely modest. No monument, said he, should 
stand over his grave, only an apple-tree, in memory of the 
three apples ; the two of Eve and Paris, which made hell out 
of earth, and that of Newton, which elevated the earth again 
into the circle of heavenly bodies.^ His son, Johann Bolyai 
(1802-1860), was educated for the army, and distinguished 
himself as a profound mathematician, an impassioned violin- 
player, and an expert fencer. He once accepted the challenge 
of thirteen officers on condition that after each duel he might 
play a piece on his violin, and he vanquished them all. 

The chief mathematical work of Wolfgang Bolyai appeared 
in two volumes, 1832-1833, entitled Tentamen juventutem 
studiosam in elementa matheseos puree . . . introducendi. It 
is followed by an appendix composed by his son Johann on 
The Science Absolute of Space. Its twenty-six pages make the 
name of Johann Bolyai immortal. He published nothing else, 
but he left behind one thousand pages of manuscript which 
have never been read by a competent mathematician ! His 
father seems to have been the only person in Hungary who 
really appreciated the merits of his son's work. For thirty- 
five years this appendix, as also Lobatchewsky's researches, 
remained in almost entire oblivion. Finally Eichard Baltzer 
of the University of Giessen, in 1867, called attention to the 
wonderful researches. Johann Bolyai's Science Absolute of 



8YKTHETIG GEOMET&Y. 303 

Space and Lobatchewsky's Oeometricdl Researches on the 
Theory of Parallels (1840) were rendered easily accessible to 
American readers by translations into English made in 1891 
by George Bruce Halsted of the University of Texas. 

The Eussian and Hungarian mathematicians were not the 
only ones to whom pangeometry suggested itself. A copy of 
the Tentamen reached Gauss, the elder Bolyai's former room- 
mate at Gottingen, and this Nestor of Grerman mathematicians 
was surprised to discover in it worked out what he himself 
had begun long before, only to leave it after him in his 
papers. As early as 1792 he had started on researches of that 
character. His letters show that in 1799 he was trying to 
prove a priori the reality of Euclid's system ; but some time 
within the next thirty years he arrived at the conclusion 
reached by Lobatchewsky and Bolyai. In 1829 he wrote to 
Bessel, stating that his ''conviction that we cannot found 
geometry completely a priori has become, if possible, still 
firmer," and that "if number is merely a product of our 
mind, space has also a reality beyond our mind of which we 
cannot fully foreordain the laws a priori" The term rwm- 
Euclidean geometry is due to Gauss. It has recently been 
brought to notice that Qeronimo Saccheriy a Jesuit father of 
Milan, in 1733 anticipated Lobatchewsky's doctrine of the 
parallel angle. Moreover, G. B. Halsted has pointed out 
that in 1766 Lambert wrote a paper " Zur Theorie der Parallel- 
linien," published in the Leipziger Magazin /Ur reine und 
angewandte Mathematiky 1786, in which: (1) The failure of 
the parallel-axiom in surface-spherics gives a geometry with 
angle-sum > 2 right angles; (2) In order to make intuitive 
a geometry with angle-sum < 2 right angles we need the aid 
of an "imaginary sphere" (pseudo-sphere); (3) In a space 
with the angle-sum differing from 2 right angles, there is 
an absolute measure (Bolyai's natural unit for length). 



804 A HISTORY OF MATHEMATICS. 

In 1854, nearly twenty years later, Gauss heard from his 
pupil, RiemanUy a marvellous dissertation carrying the dis- 
cussion one step further by developing the notion of ri-ply 
extended magnitude, and the measure-relations of which a 
manifoldness of n dimensions is capable, on the assumption 
that every line may be measured by every other. Kiemann 
applied his ideas to space. He taught us to distinguish 
between "unboundedness'' and "infinite extent." According 
to him we have in our mind a more general notion of space, 
1.6. a notion of non-Euclidean space ; but we learn by experience 
that our physical space is, if not exactly, at least to high 
degree of approximation, Euclidean space. Eiemann's pro- 
found dissertation was not published until 1867, when it 
appeared in the Oottingen Abhandlungen, Before this the 
idea of n dimensions had suggested itself under various 
aspects to Lagrange, Pliicker, and H. Grassmann. About the 
same time with Riemann's paper, others were published from 
the pens of Helmholtz and Beltrami. These contributed pow- 
erfully to the victory of logic over excessive empiricism. This 
period marks the beginning of lively discussions upon this sub- 
ject. Some writers — Bellavitis, for example — were able to 
see in non-Euclidean geometry and n-dimensional space noth- 
ing but huge caricatures, or diseased outgrowths of mathe- 
matics. Helmholtz's article was entitled ThatscLchen, welche 
der Oeometrie zu Grunde liegen, 1868, and contained many of 
the ideas of Riemann. Helmholtz popularised the subject in 
lectures, and in articles for various magazines. 

Eugenio Beltrami, born at Cremona, Italy, in 1835, and now 
professor at Rome, wrote the classical paper Saggio di inter- 
pretazione della geometria iion-euclidea {Giom. di Matem., 6), 
which is analytical (and, like several other papers, should be 
mentioned elsewhere were we to adhere to a strict separation 
between synthesis and analysis). He reached the brilliant 



8YKTHETIC 6BOMET&Y. 805 

and surprising conclusion that the theorems of non-Euclidean 
geometry find their realisation upon surfaces of constant nega- 
tive curvature. He studied, also, surfaces of constant positive 
curvature, and ended with the interesting theorem that the 
space of constant positive curvature is contained in the space 
of constant negative curvature. These researches of Beltrami, 
Helmholtz, and Biemann culminated in the conclusion that 
on surfaces of constant curvature we may have three geome- 
tries, — the non-Euclidean on a surface of constant negative 
curvature, the spherical on a surface of constant positive cur- 
vature, and the Euclidean geometry on a surface of zero curva- 
ture. The three geometries do not contradict each other, but 
are members of a system, — a geometrical trinity. The ideas 
of hyper-space were brilliantly expounded and popularised in 
England by Clifford. 

William Kingdon Clifford (1845-1879) was bom at Exeter, 
educated at Trinity College, Cambridge, and from 1871 until 
his death professor of applied mathematics in University Col- 
lege, London. His premature death left incomplete several 
brilliant researches which he had entered upon. Among these 
are his paper On Classification of Loci and his Theory of 
Graphs, He wrote articles On tlie Canonical Form and 
Dissection of a Biemann^s Surface, on Biquaiemions, and 
an incomplete work on the Elements of Dynamic. The 
theory of polars of curves and surfaces was generalised by 
him and by Reye. His classification of loci, 1878, being a 
general study of curves, was an introduction to the study 
of n-dimensional space in a direction mainly projective. 
This study has been continued since chiefly by G. Veronese 
of Padua, C. Segre of Turin, E. Bertini, F. Aschieri, P. Del 
Pezzo of Naples. 

Beltrami's researches on non-Euclidean geometry were fol- 
lowed, in 1871y by important investigations of Felix Klein, 



806 A HISTORY OF MATHEMATICS. 

resting upon Cayley's Sixth Memoir on QuanticSf 1859. The 
question whether it is not possible to so express the metrical 
properties of figures that they will not vary by projection (or 
linear transformation) had been solved for special projections 
by Chasles, Poncelet, and E. Laguerre (1834-1886) of Paris, 
but it remained for Cayley to give a general solution by defin- 
ing the distance between two points as an arbitrary constant 
multiplied by the logarithm of the anharmonic ratio in which 
the line joining the two points is divided by the fundamental 
quadric. Enlarging upon this notion, Klein showed the inde- 
pendence of projective geometry from the parallel-axiom, and 
by properly choosing the law of the measurement of distance 
deduced from projective geometry the spherical, Euclidean, 
and pseudospherical geometries, named by him respectively 
the elliptic, parabolic, and hyperbolic geometries. This sug- 
gestive investigation was followed up by numerous writers, 
particularly by G. Battaglini of Naples, E. d' Ovidio of Turin, 
R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann 
of Munich, E. Schering of Gottingen, W. Story of Clark 
University, H. Stahl of Tubingen, A. Voss of Wiirzburg, 
Homersham Cox, A. Buchheim." The geometry of n dimen- 
sions was studied along a line mainly metrical by a host of 
writers, among whom may be mentioned Simon Newcomb of 
the Johns Hopkins University, L. Schlafli of Bern, W. I. 
Stringham of the University of California, W. Killing of 
Miinster, T. Craig of the Johns Hopkins, R. Lipschitz of 
Bonn. R. S. Heath and Killing investigated the kinematics 
and mechanics of such a space. Regular solids in ri-dimen- 
sional space were studied by Stringham, Ellery W. Davis 
of the University of Nebraska, R. Hoppe of Berlin, and 
others. Stringham gave pictures of projections upon our 
space of regular solids in four dimensions, and Schlegel at 
Hagen constructed models of such projections. These are 



AKALYnO OXOMBTRY. 307 

among the most curious of a series of models published by 
L. Brill in Darmstadt. It has been pointed out that if a 
fourth dimension existed, certain motions could take place 
which we hold to be impossible. Thus Newcomb showed the 
possibility of turning a closed material shell inside out by sim- 
ple flexure without either stretching or tearing ; Klein pointed 
out that knots could not be tied; Veronese showed that a 
body could be removed from a closed room without breaking 
the walls ; G. S. Peirce proved that a body in four-fold space 
either rotates about two axes at once, or cannot rotate without 
losing one of its dimensions. 



ANALYTIC GEOMETRY. 

In the preceding chapter we endeavoured to give a flash- 
light view of the rapid advance of synthetic geometry. In 
connection with hyperspace we also mentioned analytical 
treatises. Modem synthetic and modern analytical geome- 
try have much in common, and may be grouped together 
under the common name "projective geometry." Each has 
advantages over the other. The continual direct viewing of 
figures as existing in space adds exceptional charm to the 
study of the former, but the latter has the advantage in this, 
that a well-established routine in a certain degree may outhin 
thought itself, and thereby aid original research. While in 
Germany Steiner and von Staudt developed synthetic geome- 
try, JMiicker laid the foundation of modern analytic geometry. 

Julius Pliicker (1801-1868) was born at Elberfeld, in Prus- 
sia. After studying at Bonn, Berlin, and Heidelberg, he spent 
a short time in Paris attending lectures of Monge and his 
pupils. Between 1826 and 1836 he held positions successively 
at Bonn, Berlin, and Halle. He then became professor of 



808 A HISTORY OF MATHEMATICS. 

physics at Bonn. Until 1846 his original researches were on 
geometry. In 1828 and in 1831 he published his Analfftisch- 
Oeometrische Untersuchungen in two volumes. Therein he 
adopted the abbreviated notation (used before him in a more 
restricted way by Bobillier), and avoided the tedious process 
of algebraic elimination by a geometric consideration. In the 
second volume the principle of duality is formulated analyti- 
cally. With him duality and homogeneity found expression 
already in his system of co-ordinates. The homogenous or 
tri-linear system used by him is much the same as the co-or- 
dinates of Mobius. In the identity of analytical operation 
and geometric construction Pliicker looked for the source of 
his proofs. The System der Analytischen Oeometrie, 1835, con- 
tains a complete classification of plane curves of the third 
order, based on the nature of the points at infinity. The 
Theorie der Algehraischen Curveriy 1839, contains, besides an 
enumeration of curves of the fourth order, the analytic rela- 
tions between the ordinary singularities of plane curves 
known as "Pliicker's equations," by which he was able to 
explain "Poncelet's paradox." The discovery of these rela- 
tions is, says Cayley, "the most important one beyond all 
comparison in the entire subject of modem geometry." But 
in Germany Plucker's researches met with no favour. His 
method was declared to be unproductive as compared with 
the synthetic method of Steiner and Poncelet! His rela- 
tions with Jacobi were not altogether friendly. Steiner once 
declared that he would stop writing for Crelle's Journal if 
Pliicker continued to contribute to it.* The result was that 
many of Pliicker's researches were published in foreign jour- 
nals, and that his work came to be better known in France 
and England than in his native country. The charge was 
also brought against Pliicker that, though occupying the chair 
of physics, he was no physicist. This induced him to relin- 



ANALYTIC GEOMETRY. 809 

quish mathematics, and for nearly twenty years to devote 
his energies to physics. Important discoveries on Fresnel's 
wave-surface, magnetism, spectrum-analysis were made by 
him. But towards the close of his life he returned to his 
first love, — mathematics, — and enriched it with new discov- 
eries. By considering space as made up of lines he created 
a '< new geometry of space." Regarding a right line as a 
curve involving four arbitrary parameters, one has the whole 
system of lines in space. By connecting them by a single 
relation, he got a '^ complex" of lines; by connecting them 
with a twofold relation, he got a "congruency " of lines. His 
first researches on this subject were laid before the Koyal 
Society in 1865. His further investigations thereon appeared 
in 1868 in a posthumous work entitled Neue Oeometrie dea 
Baumes gegriindet auf die BetracJUung der geraden Linie als 
Baumelementf edited by Felix Klein. Plttckei-'s analysis lacks 
the elegance found in Lagrange, Jacobi, Hesse, and Clebsch. 
For many years he had not kept up with the progress of 
geometry, so that many investigations in his last work had 
already received more general treatment on the part of others. 
The work contained, nevertheless, much that was fresh and 
original. The theory of complexes of the second degree, left 
unfinished by Pliicker, was continued by Felix Klein, who 
greatly extended and supplemented the ideas of his master. 

Ludwig Otto Hesse (1811-1874) was bom at Konigsberg, and 
studied at the university of his native place under Bessel, 
Jacobi, Richelot, and F. Neumann. Having taken the doctor's 
degree in 1840, he became docent at Konigsberg, and in 1845 
extraordinary professor there. Among his pupils at that time 
were Dur^ge, Carl Neumann, Clebsch, Kirchhoff. The Konigs- 
berg period was one of great activity for Hesse. Every new 
discovery increased his zeal for still greater achievement. 
His earliest researches were on surfaces of the second order. 



810 A HISTORY OF MATHEMATICS. 

and were partly synthetic. He solved the problem to construct 
any tenth point of such a surface when nine points are given. 
The analogous problem for a conic had been solved by Pascal 
by means of the hexagram. A difficult problem confronting 
mathematicians of this time was that of elimination. Pliicker 
had seen that the main advantage of his special method in 
analytic geometry lay in the avoidance of algebraic elimina- 
tion. Hesse, however, showed how by determinants to make 
algebraic elimination easy. In his earlier results he was 
anticipated by Sylvester, who published his dialytic method 
of elimination in 1840. These advances in algebra Hesse 
applied to the analytic study of curves of the third order. By 
linear substitutions, he reduced a form of the third degree in 
three variables to one of only four terms, and was led to an 
important determinant involving the second differential coeffi- 
cient of a form of the third degree, called the "Hessian." 
The " Hessian " plays a leading part in the theory of invari- 
ants, a subject first studied by Cay ley. Hesse showed that 
his determinant gives for every curve another curve, such that 
the double points of the first are points on the second, or 
"Hessian." Similarly for surfaces (Crelle, 1844). Many of 
the most important theorems on curves of the third order are 
due to Hesse. He determined the curve of the 14th order, 
which passes through the 56 points of contact of the 28 bi- 
tangents of a curve of the fourth order. His great memoir on 
this subject (Crelle, 1855) was published at the same time as 
was a paper by Steiner treating of the same subject. 

Hesse's income at Konigsberg had not kept pace with his 
growing reputation. Hardly was he able to support himself 
and family. In 1855 he accepted a more lucrative position at 
Halle, and in 1856 one at Heidelberg. Here he remained until 
1868, when he accepted a position at a technic school in 
Munich.^ At Heidelberg he revised and enlarged upon his 



ANALYTIC GEOMBTBY. 811 

previous researches, and published in 1861 his Vorlesungen 
iiber die AncUytische Oeometrie des Raumea, inabesondere fiber 
FUkhen 2. Ordnung, More elementary works soon followed. 
While in Heidelberg he elaborated a principle, his " Uebertra- 
gungsprincip." According to this, there corresponds to every 
X)oint in a plane a pair of points in a line, and the projective 
geometry of the plane can be carried back to the geometry of 
points in a line. 

The researches of Pliicker and Hesse were continued in Eng- 
land by Cayley, Salmon, and Sylvester. It may be premised here 
that among the early writers on analytical geometry in England 
was James Booth (1806-1878), whose chief results are embodied 
in his Treatise on Some New Oeometrical Methods; and James 
HacCollagh (1809-1846), who was professor of natural philos- 
ophy at Dublin, and made some valuable discoveries on the 
theory of quadrics. The influence of these men on the 
progress of geometry was insignificant, for the interchange of 
scientific results between different nations was not so complete 
at that time as might have been desired. In further illustra- 
tion of this, we mention that Chasles in France elaborated 
subjects which had previously been disposed of by Steiner in 
Germany, and Steiner published researches which had been 
given by Cayley, Sylvester, and Salmon nearly five years 
earlier. Cayley and Salmon in 1849 determined the straight 
lines in a cubic surface, and studied its principal properties, 
while Sylvester in 1851 discovered the pentahedron of such a 
surface. Cayley extended Pliicker's equations to curves of 
higher singularities. Cayley's own investigations, and those 
of M. Nother of Erlangen, G. H. Halphen (1844-1889) of the 
Polytechnic School in Paris, De La Goum^rie of Paris, A. 
Brill of Tiibingen, lead to the conclusion that each higher sin- 
gularity of a curve is equivalent to a certain number of simple 
singularities, — the node, the ordinary cusp, the double tangent. 



812 A HISTOBY OF MATHEMATICS. 

• 

and the inflection. Sylvester studied the "twisted Cartesian/' 
a curve of the fourth order. Salmon helped powerfully 
towards the spreading of a knowledge of the new algebraic and 
geometric methods by the publication of an excellent series of 
text-books {Conic SectConSj Modern Higher Algebra, Higher 
Plane Cwves, Geometry of Three Dimensions), which have 
been placed within easy reach of German readers by a free 
translation, with additions, made by Wilhelm Fiedler of the 
Polytechnicum in Ziirich. The next great worker in the field 
of analytic geometry was Clebsch. 

Rudolf Friedrich Alfred Clebsch (1833-1872) was born at 
Konigsberg in Prussia, studied at the university of that place 
under Hesse, Richelot, F. Neumann. From 1858 to 1863 he 
held the chair of theoretical mechanics at the Polytechnicum 
in Carlsruhe. The study of Salmon's works led him into 
algebra and geometry. In 1863 he accepted a position at the 
University of Giesen, where he worked in conjunction with 
Paul Gordan (now of Erlangen). In 1868 Clebsch went to 
Gottingen, and remained there until his death. He worked 
successively at the following subjects : Mathematical physics, 
the calculus of variations and partial differential equations of 
the first order, the general theory of curves and surfaces, 
Abelian functions and their use in geometry, the theory of 
invariants, and " Flachenabbildung." " He proved theorems 
on the pentahedron enunciated by Sylvester and Steiner; he 
made systematic use of "deficiency" (Oeschlecht) as a funda- 
mental principle in the classification of algebraic curves. The 
notion of deficiency was known before him to Abel and Rie- 
mann. At the beginning of his career, Clebsch had shown 
how elliptic functions could be advantageously applied to 
Malfatti's problem. The idea involved therein, viz. the use 
of higher transcendentals in the study of geometry, led him 
to his greatest discoveries. Not only did he apply Abelian 



ANALYTIC GEOMETRY. 313 

functions to geometry, but conversely, he drew geometry into 
the service of Abelian functions. 

Clebsch made liberal use of determinants. His study of 
curves and surfaces began with the determination of the points 
of contact of lines which meet a surface in four consecutive 
points. Salmon had proved that these points lie on the inter- 
section of the surface with a derived surface of the degree 
lln — 24, but his solution was given in inconvenient form. 
Clebsch's investigation thereon is a most beautiful piece of 
analysis. 

The representation of one surface upon another {Fldchenab- 
bildung), so that they have a (1, 1) correspondence, was 
thoroughly studied for the first time by Clebsch. The repre- 
sentation of a sphere on a plane is an old problem which 
drew the attention of Ptolemaeus, Gerard Mercator, Lambert, 
Gauss, Lagrange. Its importance in the construction of maps 
is obvious. Gauss was the first to represent a surface upon 
another with a view of more easily arriving at its properties. 
Pliicker, Chasles, Cayley, thus represented on a plane the 
geometry of quadric surfaces ; Clebsch and Cremona, that of 
cubic surfaces. Other surfaces have been studied in the same 
way by recent writers, particularly M. Nother of Erlangen, 
Armenante, Felix Klein, Korndorfer, Caporali, H. G. Zeuthen 
of Copenhagen. A fundamental question which has as yet 
received only a partial answer is this : What surfaces can be 
represented by a (1, 1) correspondence upon a given surface ? 
This and the analogous question for curves was studied by 
Clebsch. Higher correspondences between surfaces have been 
investigated by Cayley and Nother. The theory of surfaces 
has been studied also by Joseph Alfred Serret (1819-1885), pro- 
fessor at the Sorbonne in Paris, Jean Gaston Darboox of Paris, 
John Casey of Dublin (died 1891), W, R. W. Roberts of Dub- 
lin, H. Schrdter (1829-1892) of Breslau. Surfaces of the 



814 A HISTORY OF MATHEMATICS. 

fourth order were investigated by Kummer, and Fresnel's 
wave-surface, studied by Hamilton, is a particular case of 
Rummer's quartic surface, with sixteen canonical points and 
sixteen singular tangent planes.* 

The infinitesimal calculus was first applied to the determi- 
nation of the measure of curvature of surfaces by LagrangCi 
Euler, and Meunier (1754-1793) of Paris. Then followed the 
researches of Monge and Dupin, but they were eclipsed by 
the work of Gauss, who disposed of this difficult subject in a 
way that opened new vistas to geometricians. His treat- 
ment is embodied in the Disquisitiones generales circa super- 
Jicies cu'i'vas (1827) and Unterstichungen iiber gegenstdnde der 
MTieren Geoddsie of 1843 and 1846. He defined the measure 
of curvature at a point to be the reciprocal of the product 
of the two principal radii of curvature at that point. From 
this flows the theorem of Johann August Orunert (1797-1872 ; 
professor in Greifswald), that the arithmetical mean of the 
radii of curvature of all normal sections through a point is the 
radius of a sphere which has the same measure of curvature 
as has the surface at that point. Gauss's deduction of the 
formula of curvature was simplified through the use of deter- 
minants by Heinrich Richard Baltzer (1818-1887) of Giessen.* 
Gauss obtained an interesting theorem that if one surface be 
developed (abgeicicJcelt) upon another, the measure of curva^ 
ture remains unaltered at each point. The question whether 
two surfaces having the same curvature in corresponding 
points can be unwound, one upon tlie other, was answered 
by F. Minding in the affirmative only when the curvature is 
constant. The case of variable curvature is difficult, and was 
studied by Minding, J. Liouville (1806-1882) of the Poly- 
technic School in Paris, Ossian Bonnet of Paris (died 1892). 
Gauss's measure of curvature, expressed as a function of cur- 
vilinear co-ordinates, gave an impetus to the study of differ- 



• ALOEBRA. 815 

ential-invariants, or differential-parameters, which have been 
investigated by Jacobi, G. Neumann, Sir James Cockle, 
Halphen, and elaborated into a general theory by Beltrami, 
S. Lie, and others. Beltrami showed also the connection 
between the measure of curvature and the geometric axioms. 

Various researches have been brought under the head of 
^'analysis situs." The subject was first investigated by 
Leibniz, and was later treated by Gauss, whose theory of 
knots ( VerscTUingungen) has been employed recently by J. B. 
Listing, 0. Simony, F. Dingeldey, and others in their "topo- 
logic studies." Tait was led to the study of knots by Sir 
William Thomson's theory of vortex atoms. In the hands 
of Kiemann the analysis situs had for its object the deter- 
mination of what remains unchanged under transformations 
brought about by a combination of infinitesimal distortions. 
In continuation of his work, Walter Dyck of Munich wrote on 
the analysis situs of three-dimensional spaces. 

Of geometrical text-books not yet mentioned, reference 
should be made to Alfred Clebsch'a Vorlesungen iiber Oeonie- 
trie, edited by Ferdinand Lindemann, now of Munich ; Frost's 
Solid Oeometry; Durege'a Ebene Curven dritter Ordnung, 



ALGEBRA. 

The progress of algebra in recent times may be considered 
under three principal heads : the study of fundamental laws 
and the birth of new algebras, the growth of the theory of 
equations, and the development of what is called modem 
higher algebra. 

We have already spoken of George Peacock and D. F. 
Gregory in connection with the fundamental laws of algebra. 
Much was done in this line by De Morgan. 



816 A HISTORY OF MATHEMATICS. • 

Augustus De Morgan (1806-1871) was bom at Madura (Ma- 
dras), and educated at Trinity College, Cambridge. His scru- 
ples about the doctrines of the established church prevented 
him from proceeding to the M.A. degree, and from sitting 
for a fellowship. In 1828 he became professor at the newly 
established University of London, and taught there until 
1867, except for five years, from 1831-1835. De Morgan was 
a unique, manly character, and preeminent as a teacher. The 
value of his original work lies not so much in increasing our 
stock of mathematical knowledge as in putting it all upon a 
thoroughly logical basis. He felt keenly the lack of close 
reasoning in mathematics as he received it. He said once: 
"We know that mathematicians care no more for logic than 
logicians for mathematics. The two eyes of exact science are 
mathematics and logic: the mathematical sect puts out the 
logical eye, the logical sect puts out the mathematical eye; 
each believing that it can see better with one eye than with 
two." De Morgan saw with both eyes. He analysed logic 
mathematically, and studied the logical analysis of the laws, 
symbols, and operations of mathematics ; he wrote a Formal 
Logic as well as a Double Algebra, and corresponded both with 
Sir William Hamilton, the metaphysician, and Sir William 
Rowan Hamilton, the mathematician. Few contemporaries 
were as profoundly read in the history of mathematics as 
was De Morgan. No subject was too insignificant to receive 
his attention. The authorship of "Cocker's Arithmetic" and 
the work of circle-squarers was investigated as minutely as was 
the history of the invention of the calculus. Numerous arti- 
cles of his lie scattered in the volumes of the Penny and Eng- 
lish Cyclopaedias. His Differential Calculus, 1842, is still a 
standard work, and contains much that is original with the 
author. For the Encydopcedia Metropolitana he wrote on the 
calculus of functions (giving principles of symbolic reasoning) 



ALGEBRA. 817 

and on the theory of probability. Celebrated is his Budget of 
Paradoxes, 1872. He published memoirs " On the Foundation 
of Algebra" {Trans, of Cam. Phil Soc,, 1841, 1842, 1844, and 
1847). 

In Germany symbolical algebra was studied by Martin Ohm, 
who wrote a System der Matkematik in 1822. The ideas of 
Peacock and De Morgan recognise the possibility of algebras 
which differ from ordinary algebra. Such algebras were 
indeed not slow in forthcoming, but, like non-Euclidean 
geometry, some of them were slow in finding recognition. 
This is true of Grassmann's, Bellavitis's, and Peirce's dis- 
coyeries, but Hamilton's quaternions met with immediate 
appreciation in England. These algebras offer a geometrical 
interpretation of imaginaries. During the times of Descartes, 
Newton, and Euler, we have seen the negative and the imagi- 
nary, V— 1, accepted as numbers, but the latter was still 
regarded as an algebraic fiction. The first to give it a geomet- 
ric picture, analogous to the geometric interpretation of the 
negative, was H. Kiihn, a teacher in Danzig, in a publication of 
1750-1751. He represented aV— 1 by a line perpendicular 
to the line a, and equal to a in length, and construed V— 1 as 
the mean proportional between -f 1 and — 1. This same idea 
was developed further, so as to give a geometric interpretation 
of a'\' V— 6, by Jean-Robert Argand (17G8-?) of Geneva, 
in a remarkable Essai (1806) ."^ The writings of K(ihn and 
Argand were little noticed, and it remained for Gauss to break 
down the last opposition to the imaginary. He introduced i as 
an indej)endent unit co-ordinate to 1, and a -f i& as a '^ complex 
number." The connection between complex numbers and 
points on a plane, though artificial, constituted a powerful 
aid in the further study of symbolic algebra. The mind 
required a visual representation to aid it. The notion of 
what we now call vectors was growing upon mathematicians, 



318 A HISTORY OF MATHEMATICS. 

and the geometric addition of vectors in space was discovered 
independently by Hamilton^ Grassmann, and others, about the 
same time. 

WiUiam Rowan Hamilton (1805-1866) was bom of Scotch 
parents in Dublin. ELis early education; carried on at home, 
was mainly in languages. At the age of thirteen he is said to 
have been familiar with as many languages as he had lived 
years. About this time he came across a copy of Newton's 
Universal Arithmetic, After reading that, he took up succes- 
sively analytical geometry, the calculus, Newton's PrindpiOj 
Laplace's Micanique Cdeste, At the age of eighteen he 
published a paper correcting a mistake in Laplace's work. In 
1824 he entered Trinity College, Dublin, and in 1827, while he 
was still an undergraduate, he was appointed to the chair of 
astronomy. His early papers were on optics. In 1832 he 
predicted conical refraction, a discovery by aid of mathe- 
matics which ranks with the discovery of Neptune by 
Le Verrier and Adams. Then followed papers on the Prin- 
ciple of Varying Action (1827) and a general method of 
dynamics (1834-1835). He wrote also on the solution of 
equations of the fifth degree, the hodograph, fluctuating 
functions, the numerical solution of differential equations. 

The capital discovery of Hamilton is his quaternions, in 
which his study of algebra culminated. In 1835 he published 
in the Traiisactions of the Royal Irish Academy his Theory of 
Algebraic Couples. He regarded algebra " as being no mere 
art, nor language, nor primarily a science of quantity, but 
rather as the science of order of progression." Time appeared 
to him as the picture of such a progression. Hence his defini- 
tion of algebra as "the science of pure time." It was the 
subject of years' meditation for him to determine what he 
should regard as the product of each pair of a system of per- 
pendicular directed lines. At last, on the 16th of October, 



ALOBBRA. 819 

1843, while walking with his wife one evening, along the 
Koyal Canal in Dublin, the discovery of quaternions flashed 
upon him, and he then engraved with his knife on a stone in 
Brougham Bridge the fundamental formula i* = j* = A:* = yfe = 
— 1. At the general meeting of the Irish Academy, a month 
later, he made the first communication on quaternions. An 
account of the discovery was given the following year in the 
Philosophical Magazine, Hamilton displayed wonderful fer- 
tility in their development. His Lectures on Quaternions, 
delivered in Dublin, were printed in 1852. His Elements of 
Quaternions appeared in 1866. Quaternions were greatly 
admired in Englaud from the start, but on the Continent they 
received less attenttion. P. O. Tait's Elementary Treatise 
helped powerfully to spread a knowledge of them in England. 
Cayley, Clifford, and Tait advanced the subject somewhat by 
original contributions. But there has been little progress in 
recent years, except that made by Sylvester in the solution of 
quaternion equations, nor has the application of quaternions 
to physics been as extended as was predicted. The change 
in notation made in France by Hoiiel and by Laisant has been 
considered in England as a wrong step, but the true cause for 
the lack of progress is perhaps more deep-seated. There is 
indeed great doubt as to whether the quaternionic product can 
claim a necessary and fundamental place in a system of vector 
analysis. Physicists claim that there is a loss of naturalness 
in taking the square of a vector to be negative. In order to 
meet more adequately their wants, J. W. Oihbs of Yale Uni- 
versity and A, MacfarUxne of the University of Texas, have 
each suggested an algebra of vectors with a new notation. 
Each gives a definition of his own for the product of two 
vectors, but in such a way that the square of a vector is 
positive. A third system of vector analysis has been used by 
Oliver Heaviside in his electrical researches. 



320 A HISTORY OF MATHEMATICS. 

Hermann Grassmann (1809-1877) was born at Stettin, 
attended a gymnasium at his native place (where his &ther 
was teacher of mathematics and physics), and studied theology 
in Berlin for three years. In 1834 he succeeded Steiner as 
teacher of mathematics in an industrial school in Berlin, but 
returned to Stettin in 1836 to assume the duties of teacher of 
mathematics, the sciences, and of religion in a school there.'^ 
Up to this time his knowledge of mathematics was pretty 
much confined to what he had learned from his father, who 
had written two books on " Raumlehre " and " Grossenlehre." 
But now he made his acquaintance with the works of Lacroix, 
Lagrange, and Laplace. He noticed that Laplace's results 
could be reached in a shorter way by some new ideas advanced 
in his father's books, and he proceeded to elaborate this 
abridged method, and to apply it in the study of tides. He 
was thus led to a new geometric analysis. In 1840 he had 
made considerable progress in its development, but a new 
book of Schleiermacher drew him again to theology. In 
1842 he resumed mathematical research, and becoming thor- 
oughly convinced of the importance of his new analysis, 
decided to devote himself to it. It now became his ambition 
to secure a mathematical chair at a university, but in this he 
never succeeded. In 1844 appeared his great classical work, 
the Lineale Ausdehnungslehre, which was full of new and 
strange matter, and so general, abstract, and out of fashion in 
its mode of exposition, that it could hardly have had less 
influence on European mathematics during its first twenty 
years, had it been published in China. Gauss, Grunert, and 
Mobius glanced over it, praised it, but complained of the 
strange terminology and its " philosophische Allgemeinheit." 
Eight years afterwards, Bretsclineider of Gotha was said to be 
the only man who had read it through. An article in Crelle^s 
Joumalj in which Grassmann eclipsed the geometers of that 



ALGEBRA. 321 

time by constructing, with aid of his method; geometrically 
any algebraic curve, remained again unnoticed. Need we mar- 
vel if Grassmann turned his attention to other subjects, — to 
Schleiermacher's philosophy, to politics, to philology ? Still, 
articles by him continued to appear in Crelle^s Journal^ and in 
1862 came out the second part of his Ausdehnungslehre, It 
was intended to show better than the first part the broad 
scope of the Ausdehnungslehre, by considering not only geo- 
metric applications, but by treating also of algebraic functions, 
infinite series, and the differential and integral calculus. But 
the second part was no more appreciated than the first. At the 
age of fifty-three, this wonderful man, with heavy heart, gave 
up mathematics, and directed his energies to the study of Sans- 
krit, achieving in philology results which were better appreci- 
ated, and which vie in splendour with those in mathematics. 

Common to the Ausdehnungslehre and to quaternions are geo- 
metric addition, the function of two vectors represented in qua- 
ternions by SaP and VaP, and the linear vector functions. The 
quaternion is peculiar to Hamilton, while with Grassmann we 
find in addition to the algebra of vectors a geometrical algebra 
of wide application, and resembling Mobius's BarycerUrische 
Calculy in which the point is the fundamental element. Grass- 
mann developed the idea of the "external product," the "inter- 
nal product," and the " open product." The last we now call 
a matrix. His Ausdehnungslehre has very great extension, 
having no limitation to any particular number of dimen- 
sions. Only in recent years has the wonderful richness of 
his discoveries begun to be appreciated. A second edition of 
the Ausdehnungslehre of 1844 was printed in 1877. C. S. 
Peirce gave a representation of Grassmann's system in the 
logical notation, and E. W. Hyde of the University of Cin- 
cinnati wrote the first text-book on Grassmann's calculus in 
the English langiiage. 



822 A mSTOBY OF MATHEMATICS. 

Discoveries of less value, which in part covered those of 
Orassmann and Hamilton, were made by SairU-Venant (1797- 
1886), who described the multiplication of vectors, and the 
addition of vectors and oriented areas; by Cauchy, whose 
"clefs alg^briques" were units subject 'to combinatorial mul-' 
tiplication, and were applied by the author to the theory of 
elimination in the same way as had been done earlier by 
Orassmann; by Justus Bellavitis (1803-1880), who published 
in 1835 and 1837 in the Anndli delle Scienze his calculus of 
sequipoUences. Bellavitis, for many years professor at Padua^ 
was a self-taught mathematician of much power, who in his 
thirty-eighth year laid down a city office in his native place, 
Bassano, that he might give his time to science.® 

The first impression of Orassmann's ideas is marked in the 
writings of Hermann Hankel (1839-1873), who published in 
1867 his Vorlesungen ilber die Complexen ZaJUen, Hankel, 
then decent in Leipzig, had been in correspondence with 
Grassmann. The "alternate numbers'' of Hankel are sub- 
ject to his law of combinatorial multiplication. In consider- 
ing the foundations of algebra Hankel affirms the principle of 
the permanence of formal laws previously enunciated incom- 
pletely by Peacock. Hankel was a close student of mathe- 
matical history, and left behind an unfinished work thereon. 
Before his death he was professor at Tubingen. His Com- 
plexe Zahlen was at first little read, and we must turn to 
Victor Schlegel of Hagen as the successful interpreter of Grass- 
mann. Schlegel was at one time a young colleague of Grass- 
mann at the Marienstifts-Gymnasium in Stettin. Encouraged 
by Clebsch, Schlegel wrote a System der Raumlekre which 
explained the essential conceptions and operations of the 
Ausdehnungslehre. 

Multiple algebra was powerfully advanced by Peirce, whose 
theory is not geometrical, as are those of Hamilton and Grass- 



ALGEBBA. 828 

mann. Benjamin Peirce (1809-1880) was bom at Salem; Mass., 
and graduated at Harvard College, haying as undergraduate 
carried the study of mathematics far beyond the limits of the 
college course.* When Bowditch was preparing his transla- 
tion and commentary of the M4canique C4leste, young Peirce 
helped in reading the proof-sheets. He was made professor at 
Harvard in 1833, a position which he retained until his death. 
For some years he was in charge of the Nautical Almanac 
and superintendent of the United States Coast Survey. He 
published a series of college text-books on mathematics, an 
Analytical Mechanics, 1855, and calculated, together with Sears 
C. Walker of Washington, the orbit of Neptune. Profound 
are his researches on Linear Associative Algebra. The first of 
several papers thereon was read at the first meeting of the 
American Association for the Advancement of Science in 1864. 
Lithographed copies of a memoir were distributed among friends 
in 1870, but so small seemed to be the interest taken in this 
subject that the memoir was not printed until 1881 (Am. Jour. 
Math.y Vol. IV., No. 2). Peirce works out the multiplication 
tables, first of single algebras, then of double algebras, and so 
on up to sextuple, making in all 162 algebras, which he shows 
to be possible on the consideration of symbols A, B, etc., 
which are linear functions of a determinate number of letters 
or units t, j, k, I, etc., with coefficients which are ordinary ana- 
lytical magnitudes, real or imaginary, — the letters i, J, etc., 
being such that every binary combination i*, y, ji, etc., is equal 
to a linear function of the letters, but under the restriction of 
satisfying the associative law.** Charles S. Peirce, a son of 
Benjamin Peirce, and one of the foremost writers on mathe- 
matical logic, showed that these algebras were all defective 
forms of quadrate algebras which he had previously discovered 
by logical analysis, and for which he had devised a simple 
notation. Of these quadrate algebras quaternions is a simple 



824 A HISTORY OF MATHEMATICS. 

example ; nonions is another. C. S. Peirce showed that of all 
linear associative algebras there are only three in which divis- 
ion is unambiguous. These are ordinary single algebra, 
ordinary double algebra, and quaternions, from which the 
imaginary scalar is excluded. He showed that his father's 
algebras are operational and matricular. Lectures on multiple 
algebra were delivered by J. J. Sylvester at the Johns Hopkins 
University, and published in various journals. They treat 
largely of the algebra of matrices. The theory of matrices 
was developed as early as 1858 by Cayley in an important 
memoir which, in the opinion of Sylvester, ushered in the 
reign of Algebra the Second. Clifford, Sylvester, H. Taber, 
C. H. Chapman, carried the investigations much further. The 
originator of matrices is really Hamilton, but his theory, pub- 
lished in his Lectures on Quaternions, is less general than that 
of Cayley. The latter makes no reference to Hamilton. 

The theory of determinants''* was studied by Hoen^ Wronski 
in Italy and J. Binet in France ; but they were forestalled by 
the great master of this subject, Cauchy. In a paper {Jour, 
de Vecole Polyt, IX., 16) Cauchy developed several general 
theorems. He introduced the name determinant, a term 
previously used by Gauss in the functions considered by him. 
In 1826 Jacobi began using this calculus, and he gave brilliant 
proof of its power. In 1841 he wrote extended memoirs on 
determinants in Crelle^s Journal, which rendered the theory 
easily accessible. In England the study of linear transforma- 
tions of quantics gave a powerful impulse. Cayley developed 
skew-determinants and Pfaffians, and introduced the use of 
determinant brackets, or the familiar pair of upright lines. 
More recent researches on determinants appertain to special 
forms. "Continuants'' are due to Sylvester; " alternants,'* 
originated by Cauchy, have been developed by Jacobi, N. Trudi, 
H. Nagelbach, and G. Garbieri ; " axisymmetric determinants," 



AliQEBBA. 325 

first used by Jacobi, have been studied by V. A. Lebesgue, 
Sylvester, and Hesse ; " circulants " are due to E. Catalan of 
Lifege, W. Spottiswoode (1825-1883), J. W. L. Glaisher, and 
B. F. Scott; for " centro-symmetric determinants" we are 
indebted to G. Zehfuss. E. B. Christoffel of Strassburg and 
G. Frobenius discovered the properties of " Wronskians," first 
used by Wronski. V. Nachreiner and S. GCinther, both of 
Munich, pointed out relations between determinants and con- 
tinued fractions ; Scott uses Hankers alternate numbers in his 
treatise. Text-books on determinants were written by Spot- 
tiswoode (1851), Brioschi (1854), Baltzer (1857), GCinther 
(1875), Dostor (1877), Scott (1880), Muir (1882), Hanus 
(1886). 

Modem higher algebra is especially occupied with the 
theory of linear transformations. Its development is mainly 
the work of Cayley and Sylvester. 

Arthur Cayley, born at Richmond, in Surrey, in 1821, was d. ^ < 
educated at Trinity College, Cambridge.^^ He came out Senior 
Wrangler in 1842. He then devoted some years to the study 
and practice of law. On the foundation of the Sadlerian pro- 
fessorship at Cambridge, he accepted the offer of that chair, 
thus giving up a profession promising wealth for a very 
modest provision, but which would enable him to give all his 
time to mathematics. Cayley began his mathematical publi- 
cations in the Cambridge Mathematical Journal while he was 
still an undergraduate. Some of his most brilliant discoveries 
were made during the time of his legal practice. There is 
hardly any subject in pure mathematics which the genius of 
Cayley has not enriched, but most important is his creation 
of a new branch of analysis by his theory of invariants. 
(}enns of the principle of invariants are found in the writ- 
ings of Lagrange, Gauss, and particularly of Boole, who 
showed, in 1841, that invariance is a property of discrimi- 



826 A HISTORY OF MATHEMATICS. 

nants generally^ and who applied it to the theory of orthogonal 
substitution. Cayley set himself the problem to determine 
a priori what functions of the coefficients of a given equation 
possess this property of invariance, and found, to begin with, 
in 1845; that the so-called '' hyper-determinants " possessed it. 
Boole made a number of additional discoveries. Then Syl- 
vester began his papers in the Cambridge and Dublin Mafhe- 
matical Journal on the Calculus of Forms. After this, 
discoveries followed in rapid succession. At that time Cay- 
ley and Sylvester were both residents of London, and they 
stimulated each other by frequent oral communications. It 
has often been difficult to determine how much really belongs 
to each. 

James Joseph Sylvester was born in London in 1814, and 
educated at St. Johns College, Cambridge. He came out 
Second Wrangler in 1837. His Jewish origin incapacitated 
him from taking a degree. In 1846 he became a student at 
the Inner Temple, and was called to the bar in 1850. He 
became professor of natural philosophy at University College, 
London ; then, successively, professor of mathematics at the 
University of Virginia, at the Royal Military Academy in 
Woolwich, at the Johns Hopkins University in Baltimore, 
and is, since 1883, professor of geometry at Oxford. His 
first printed paper was on FresnePs optic theory, 1837. Then 
followed his researches on invariants, the theory of equations, 
theory of partitions, multiple algebra, the theory of numbers, 
and other subjects mentioned elsewhere. About 1874 he took 
part in the development of the geometrical theory of link- 
work movements, originated by the beautiful discovery of 
A. Peaucellier, Capitaine du Genie k Nice (published in 
Nouvdles AnnaleSy 1864 and 1873), and made the subject 
of close study by A. B. Kempe. To Sylvester is ascribed the 
general statement of the theory of contravariants, the dis- 



ALGBBBA. 827 

oovery of the partial differential equations satisfied by the 
invariants and covariants of binary quantics, and the subject 
of mixed concomitants. In the American Journal of Mathe- 
maiics are memoirs on binary and ternary quantics, elaborated 
partly with aid of F. Frankliny now professor at the Johns 
Hopkins University. At Oxford, Sylvester has opened up a 
new subject, the theory of reciprocants, treating of the func- 
tions of a dependent variable y and the functions of its differ- 
ential coefficients in regard to a;, which remain unaltered by 
the interchange of x and y. This theory is more general than 
one on differential invariants by Halphen (1878), and has 
been developed further by J. Hammond of Oxford, McMahon 
of Woolwich, A. R. Forsyth of Cambridge, and others. Syl- 
vester playfully lays claim to the appellation of the Mathe- 
matical Adam, for the many names he has introduced into 
mathematics. Thus the terms invariant, discriminant, Hessian, 
Jacobian, are his. 

The great theory of invariants, developed in England mainly 
by Cayley and Sylvester, came to be studied earnestly in Ger- 
many, France, and Italy. One of the earliest in the field was 
Siegfried Heinrich Aronhold (1819-1884), who demonstrated 
the existence of invariants, S and T, of the ternary cubic. 
Hermite discovered evectants and the theorem of reciprocity 
named after him. Paul Grordan showed, with the aid of 
symbolic methods, that the number of distinct forms for a 
binary quantic is finite. Clebsch proved this to be true for 
quantics with any number of variables. A very much simpler 
proof of this was given in 1891, by David Hilbert of Konigs- 
berg. In Italy, F. Brioschi of Milan and Fadt de Bruno 
(1825-1888) contributed to the theory of invariants, the 
latter writing a text-book on binary forms, which ranks by 
the side of Salmon's treatise and those of Clebsch and 
Gordan. Among other writers on invariants are E. B. Chris- 



328 A HISTORY OF MATHEMATICS. 

toffel, Wilhelm Fiedler, P. A. McMahon, J. W. L. Glaisher of 
Cambridge, Emory McClintock of New York. McMahon dis- 
covered that the theory of semi-invariants is a part of that of 
symmetric functions. The modem higher algebra has reached 
out and indissolubly connected itself with several other 
branches of mathematics — geometry, calculus of variations, 
mechanics. Glebsch extended the theory of binary forms to 
ternary, and applied the results to geometry. Glebsch, Klein, 
Weierstrass, Burckhardt, and Bianchi have used the theory of 
invariants in hyperelliptic and Abelian functions. 

In the theory of equations Lagrange, Argand, and (xauss 
furnished proof to the important theorem that every algebraic 
equation has a real or a complex root. Abel proved rigorously 
that the general algebraic equation of the fifth or of higher 
degrees cannot be solved by radicals {CreUe, I., 1826). A modi- 
fication of Abel's proof was given by Wantzel. Before Abel, 
an Italian physician, Paolo Rvfflni (1765-1822), had printed 
proofs of the insolvability, which were criticised by his 
countryman Malfatti. Though inconclusive, Ruffini's papers 
are remarkable as containing anticipations of Cauchy's theory 
of groups/* A transcendental solution of the quintic involving 
elliptic integrals was given by Herraite (Compt. Rend., 1858, 
1865, 1866). After Hermite's first publication, Kronecker, in 
1858, in a letter to Hermite, gave a second solution in which 
was obtained a simple resolvent of the sixth degree. Jerrard, in 
his Mathematical Researches (1832-1835), reduced the quintic 
to the trinomial form by an extension of the method of 
Tschirnhausen. This important reduction had been effected 
as early as 1786 by E. S. Bring y a Swede, and brought out 
in a publication of the University of Lund. Jerrard, like 
Tschirnhausen, believed that his method furnished a general 
algebraic solution of equations of any degree. In 1836 Wil- 
liam R. Hamilton made a report on the validity of Jerrard's 



ALGEBRA. 829 

method, and showed that by his process the quiutic could be 
transformed to any one of the four trinomial forms. Ham- 
ilton defined the limits of its applicability to higher equations. 
Sylvester investigated this question, What is the lowest degree 
an equation can have in order that it may admit of being 
deprived of i consecutive terms by aid of equations not higher 
than ith degree. He carried the investigation as far as i = 8, 
and was led to a series of numbers which he named ''Hamilton's 
numbers." A transformation of equal importance to Jerrard's 
is that of Sylvester, who expressed the quintic as the sum of 
three fifth-powers. The covariants and invariants of higher 
equations have been studied much in recent years. 

Abel's proof that higher equations cannot always be solved 
algebraically led to the inquiry as to what equations of a given 
degree can be solved by radicals. Such equations are the 
ones discussed by Gauss in considering the division of the 
circle. Abel advanced one step further by proving that an 
irreducible equation can always be solved in radicals, if, of 
two of its roots, the one can be expressed rationally in terms 
of the other, provided that the degree of the equation is 
prime; if it is not prime, then the solution depends upon 
that of equations of lower degree. Through geometrical con- 
siderations, Hesse came upon algebraically solvable equations 
of the ninth degree, not included in the previous groups. 
The subject was powerfully advanced in Paris by the youthful 
Evariste OaloU (bom, 1811; killed in a duel, 1832), who 
introduced the notion of a group of substitutions. To him are 
due also some valuable results in relation to another set of 
equations, presenting themselves in the theory of elliptic 
functions, viz. the modular equations. Galois's labours gave 
birth to the important theory of substitutions, which has been 
greatly advanced by C, Jordan of Paris, J, A, Serret (1819- 
1885) of the Sorbonne in Paris, L. Kronecker (1823-1891) of 



380 A HISTORY OF MATHEMATICS. 

Berlin, Klein of Gottiugen, M. Nother of Erlangen, C. 
Hermite of Paris, A. Capelli of Naples, L. Sylow of Pried- 
richshald, E. Netto of Giessen. Netto's book, the SvbsUtUr 
tionstheorie, has been translated into English by F. N. Cole of 
the University of Michigan, who contributed to the theory. 
A simple group of 504 substitutions of nine letters, discovered 
by Cole, has been shown by E. H. Moor of the University of 
Chicago to belong to a doubly-infinite system of simple groups. 
The theory of substitutions has important applications in the 
theory of differential equations. Kronecker published, in 1882, 
his Gfrundailye einer Arithmetischen Theorie der Algebraischen 
Ghr68sen. 

Since Fourier and Budan, the solution of numerical equa- 
tions has been advanced by W, O, Homer of Bath, who gave 
an improved method of approximation {Philosophical Transac- 
tionsy 1819). Jacques Charles Franqois Sturm (1803-1855), a 
native of Geneva, Switzerland, and the successor of Poisson 
in the chair of mechanics at the Sorbonne, published in 1829 
his celebrated theorem determining the number and situation 
of roots of an equation comprised between given limits. Sturm 
tells us that his theorem stared him in the face in the midst 
of some mechanical investigations connected with the motion 
of a compound pendulum." This theorem, and Horner's 
method, offer together sure and ready means of finding the 
real roots of a numerical equation. 

The symmetric functions of the sums of powers of the roots 
of an equation, studied by Newton and Waring, was considered 
more recently by Gauss, Cayley, Sylvester, Brioschi. Cayley 
gives rules for the " weight '' and " order " of symmetric func- 
tions. 

The theory of elimination was greatly advanced by Sylves- 
ter, Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and 
Qordan. Sylvester gave the dialytic method {Philosophical 



ANALYSIS. 381 

Magazine, 1840), and in 1852 established a theorem relating 
to the expression of an eliminant as a determinant. Cayley 
made a new statement of B^zout's method of elimination and 
established a general theory of elimination (1852). 



ANALYSIS. 

Under this head we find it convenient to consider the sub- 
jects of the differential and integral calculus, the calculus of 
variations, infinite series, probability, and differential equa- 
tions. Prominent in the development of these subjects was 
Cauchy. 

Angustin-Loois Canchy^ (1789-1857) was bom in Paris, 
and received his early education from his father. Lagrange 
and Laplace, with whom the father came in frequent contact, 
foretold the future greatness of the young boy. At the 
£cole Gentrale du Panth^n he excelled in ancient classical 
studies. In 1805 he entered the Polytechnic School, and two 
years later the £cole des Ponts et Ghauss^s. Cauchy left for 
Cherbourg in 1810, in the capacity of engineer. Laplace's 
Micanique Celeste and Lagrange's Fonctions Analytiques were 
among his book companions there. Considerations of health 
induced him to return to Paris after three years. Yielding to 
the persuasions of Lagrange and Laplace, he renounced engi- 
neering in favour of pure science. We find him next holding 
a professorship at the Polytechnic School. On the expulsion 
of Charles X., and the accession to the throne of Louis Philippe 
in 1830, Cauchy, being exceedingly conscientious, found him- 
self unable to take the oath demanded of him. Being, in 
consequence, deprived of his positions, he went into volun- 
tary exile. At Fribourg in Switzerland, Cauchy resumed his 
studies, and in 1831 was induced by the king of Piedmont to 



882 A HISTORY OF MATHEMATIGS. 

accept the chair of mathematical physics, especially created 
for him at the university of Turin. In 1833 he obeyed the 
call of his exiled king, Ch^J'les X., to undertake the education 
of a grandson, the Duke of Bordeaux. This gave Cauchy an 
opportunity to visit various parts of Europe, and to learn how 
extensively his works were being read. Charles X. bestowed 
upon him the title of Baron. On his return to Paris in 1838, 
a chair in the College de France was offered to him, but the 
oath demanded of him prevented his acceptance. He was 
nominated member of the Bureau of Longitude, but declared 
ineligible by the ruling power. During the political events of 
1848 the oath was suspended, and Cauchy at last became 
professor at the Polytechnic School. On the establishment of 
the second empire, the oath was re-instated, but Cauchy and 
Arago were exempt from it. Cauchy was a man of great 
piety, and in two of his publications staunchly defended the 
Jesuits. 

Cauchy was a prolific and profound mathematician. By a 
prompt publication of his results, and the preparation of 
standard text-books, he exercised a more immediate and 
beneficial influence upon the great mass of mathematicians 
than any contemporary writer. He was one of the leaders 
in infusing rigour into analysis. His researches extended 
over the field of series, of imaginaries, theory of numbers, 
differential equations, theory of substitutions, theory of func- 
tions, determinants, mathematical astronomy, light, elasticity, 
etc., — covering pretty much the whole realm of mathematics, 
pure and applied. 

Encouraged by Laplace and Poisson, Cauchy published in 
1821 his Cours d^ Analyse de VEcole Royale Polytechnique, a 
work of great merit. Had it been studied more diligently by 
writers of text-books in England and the United States, many 
a lax and loose method of analysis hardly as yet eradicated 



ANALYSIS. 383 

fmm elementary text-books would have been discarded over 
half a century ago. Gauchy was the first to publish a 
rigorous proof of Taylor's theorem. He greatly improved 
the exposition of fundamental principles of the differential 
calculus by his mode of considering limits and his new theory 
on the continuity of functions. The method of Gauchy and 
Duhamel was accepted with favour by Hoiiel and others. In 
England special attention to the clear exposition of funda- 
mental principles was given by De Morgan. Recent American 
treatises on the calculus introduce time as an independent vari- 
able, and the allied notions of velocity and acceleration — thus 
virtually returning to the method of fluxions. 

Gauchy made some researches on the calculus of variations. 
This subject is now in its essential principles the same as 
when it came from the hands of Lagrange. Recent studies 
pertain to the variation of a double integral when the limits 
are also variable, and to variations of multiple integrals in 
general. Memoirs were published by Gauss in 1829, Poisson 
in 1831, and Ostrogradsky of St. Petersburg in 1834, without, 
however, determining in a general manner the number and 
form of the equations which must subsist at the limits in case 
of a double or triple integral In 1837 Jacobi published a 
memoir, showing that the difficult integrations demanded by 
the discussion of the second variation, by which the existence 
of a maximum or minimum can be ascertained, are included 
in the integrations of the first variation, and thus are super- 
fluous. This important theorem, presented with great brevity 
by Jacobi, was elucidated and extended by V. A. Lebesgue, 
G. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and Glebsch. An 
important memoir by Sarrus on the question of determining 
the limiting equations which must be combined with the 
indefinite equations in order to determine completely the 
maxima and minima of multiple integrals, was awarded a 



8S4 A HISTORY OF MATHEMATICS. 

prize by the French Academy in 1845, honourable mention 
being made of a paper by Delaunay. Sarrus's method was 
simplified by Gauchy. In 1852 G. Mainardi attempted to 
exhibit a new method of discriminating maxima and minima, 
and extended Jacobi's theorem to double integrals. Mainardi 
and F. Brioschi showed the value of determinants in exhibit- 
ing the terms of the second variation. In 1861 Isaac Tad- 
hunter (1820-1884) of St. John's College, Cambridge, published 
his valuable work on the History of the Progress of the Calculus 
of Variations, which contains researches of his own. In 1S66 
he published a most important research, developing the theory 
of discontinuous solutions (discussed in particular cases by 
Legendre), and doing for this subject what Sarrus had done 
for multiple integrals. 

The following are the more important authors of systematic 
treatises on the calculus of variations, and the dates of publi- 
cation: Robert Woodhouse, Fellow of Caius College, Cam- 
bridge, 1810 ; Richard Abbatt in London, 1837 ; John Hewitt 
Jellett (1817-1888), once Provost of Trinity College, Dublin, 
1850 ; G. W. Strauch in Zurich, 1849 ; Moigno and Lindelof, 
1861; Lewis Buffett Carll of Flushing in New York, 1881. 

The lectures on definite integrals, delivered by Dirichlet in 
1858, have been elaborated into a standard work by G. F. 
Meyer. The subject has been treated most exhaustively by 
D. Bierens de Haan of Leiden in his Expos4 de la tMone des 
integrals d^JinieSy Amsterdam, 1862. 

The history of infinite series illustrates vividly the salient 
feature of the new era which analysis entered upon during the 
first quarter of this century. Newton and Leibniz felt the 
necessity of inquiring into the convergence of infinite series, 
but they had no proper criteria, excepting the test advanced 
by Leibniz for alternating series. By Euler and his contem- 
poraries the formal treatment of series was greatly extended, 



ANALYSIS. 835 

while the necessity for determining the convergence was gen- 
erally lost sight of. Euler reached some very pretty results 
on infinite series, now well known, and also some very 
absurd results, now quite forgotten. The faults of his time 
found their culmination in the Combinatorial School in (Ger- 
many, which has now passed into deserved oblivion. At the 
beginning of the period now under consideration, the doubtful, 
or plainly absurd, results obtained from infinite series stimu- 
lated profounder inquiries into the validity of operations with 
them. Their actual contents came to be the primary, foi^n a 
secondary, consideration. The first important and strictly 
rigorous investigation of series was made by Gauss in con- 
nection with the hypergeometric series. The criterion devel- 
oped by him settles the question of convergence in every case 
which it is intended to cover, and thus bears the stamp of 
generality so characteristic of Gauss's writings. Owing to the 
strangeness of treatment and unusual rigour. Gauss's paper 
excited little interest among the mathematicians of that time. 
More fortunate in reaching the public was Cauchy, whose 
Analyse Alg^hrique of 1821 contains a rigorous treatment of 
series. All series whose sum does not approach a fixed limit 
as the number of terms increases indefinitely are called diver- 
gent. Like Gauss, he institutes comparisons with geometric 
series, and finds that series with positive terms are convergent 
or not, according as the nth root of the nth term, or the ratio 
of the (n-f l)th term and the nth term, is ultimately less 
or greater than unity. To reach some of the cases where 
these expressions become ultimately unity and fail, Cauchy 
established two other tests. He showed that series with neg- 
ative terms converge when the absolute values of the terms 
converge, and then deduces Leibniz's test for alternating 
series. The product of two convergent series was not found 
to be necessarily convergent. Cauchy's theorem that the 



886 A HISTORY OF HATHBMATIOS. 

product of two absolutely convergent series converges to the 
product of the sums of the two series was shown half a cen- 
tury later by F. Mortens of Graz to be still true if, of the two 
convergent series to be multiplied together, only one is abso- 
lutely convergent. 

The most outspoken critic of the old methods in series 
was Abel. His letter to his friend Holmboe (1826) contains 
severe criticisms. It is very interesting reading, even to 
modern students. In his demonstration of the binomial theo- 
rem he established the theorem that if two series and their 
product series are all convergent, then the product series will 
converge towards the product of the sums of the two given 
series. This remarkable result would dispose of the whole 
problem of multiplication of series if we had a universal 
practical criterion of convergency for semi-convergent series. 
Since we do not possess such a criterion, theorems have been 
recently established by A. Pringsheim of Munich and A. Voss 
of Wiirzburg which remove in certain cases the necessity of 
applying tests of convergency to the product series by the 
application of tests to easier related expressions. Pringsheim 
reaches the following interesting conclusions: The product 
of two semi-convergent series can never converge absolutely, 
but a semi-convergent series, or even a divergent series, multi- 
plied by an absolutely convergent series, may yield an abso- 
lutely convergent product. 

The researches of Abel and Cauchy caused a considerable 
stir. We are told that after a scientific meeting in which 
Cauchy had presented his first researches on series, Laplace 
hastened home and remained there in seclusion until he had 
examined the series in his Micanique CSleste. Luckily, every 
one was found to be convergent ! We must not conclude, 
however, that the new ideas at once displaced the old. On tlie 
contrary, the new views were generally accepted only after a 



ANALYSIS. 887 

gevere and long struggle. As late as 1844 De Morgan began a 
paper on " divergent series " in this style : " I believe it will 
be generally admitted that the heading of this paper describes 
the only subject yet remaining^ of an elementary character, on 
which a serious schism exists among mathematicians as to the 
absolute correctness or incorrectness of results.'' 

First in time in the evolution of more delicate criteria of 
convergence and divergence come the researches of Josef Lud- 
wig Baabe (Crelle, Vol. IX.); then follow those of De Morgan 
as given in his calculus. De Morgan established the loga- 
rithmic criteria which were discovered in part independently 
by J. Bertrand. The forms of these criteria, as given by 
Bertrand and by Ossian Bonnet, are more convenient than 
De Morgan's. It appears from Abel's posthumous papers 
that he had anticipated the above-named writers in estab- 
lishing logarithmic criteria. It was the opinion of Bonnet 
that the logarithmic criteria never fail ; but Du Bois-Rey- 
mond and Pringsheim have each discovered series demon- 
strably convergent in which these criteria fail to determine 
the convergence. The criteria thus far alluded to have 
been called by Pringsheim special criteria, because they all 
depend upon a comparison of the nth term of the series 
with special functions a*, n", n(logn)", etc. Among the 
first to suggest general criteria, and to consider the subject 
from a still wider point of view, culminating in a regular 
mathematical theory, was Kummer. He established a theorem 
yielding a test consisting of two parts, the first part of 
which was afterwards found to be superfluous. The study 
of general criteria was continued by U. Dini of Pisa, Paul 
Du Bois-Reymond, G. Kohn of Minden, and Pringsheim. 
Du Bois-Reymond divides criteria into two classes: criteria 
of the first kind and criteria of the second kindj according as 
the general nth term, or the ratio of the (n-(-l)th term and 



888 A HISTORY OF MATHEMATICS. 

the nth term, is made the basis of research. Kummer's is a 
criterion of the second kind. A criterion of the first kind, 
analogous to this, was invented by Pringsheim. From the 
general criteria established by Du Bois-Beymond and Prings- 
heim respectively, all the special criteria can be derived. The 
theory of Pringsheim is very complete, and offers, in addition 
to the criteria of the first kind and second kind, entirely new 
criteria of a third kind, and also generalised criteria of the 
second kind, which apply, however, only to series with never 
increasing terms. Those of the third kind rest mainly on the 
consideration of the limit of the difference either of consecu- 
tive terms or of their reciprocals. In the generalised criteria 
of the second kind he does not consider the ratio of two con- 
secutive terms, but the ratio of any two terms however far 
apart, and deduces, among others, two criteria previously given 
by Kohn and Ermakoff respectively. 

Difficult questions arose in the study of Fourier's series.^ 
Cauchy was the first who felt the necessity of inquiring into 
its convergence. But his mode of proceeding was found 
by Dirichlet to be unsatisfactory. Dirichlet made the first 
thorough researches on this subject (Crelley Vol. IV.). They 
culminate in the result that whenever the function does not 
become infinite, does not have an infinite number of dis- 
continuities, and does not possess an infinite number of 
maxima and minima, then Fourier's series converges toward 
the value of that function at all places, except points of 
discontinuity, and there it converges toward the mean of 
the two boundary values. Schlafli of Bern and Du Bois- 
Reymond expressed doubts as to the correctness of the mean 
value, which were, however, not well founded. Dirichlet's 
conditions are sufficient, but not necessary. Lipschitz, of 
Bonn, proved that Fourier's series still represents the func- 
tion when the number of discontinuities is infinite, and 



ANALYSIS. 339 

established a condition on which it represents a function 
having an infinite number of maxima and minimsi. Dirich- 
let's belief that all continuous functions can be represented 
by Fourier's series at all points was shared by Kiemann and 
H. Hankel, but was proved to be false by Du Bois-Beymond 
and H. A. Schwarz. 

Kiemann inquired what properties a function must have, 
so that there may be a trigonometric series which, whenever 
it is convergent, converges toward the value of the function. 
He found necessary and sufficient conditions for this. They 
do not decide, however, whether such a series actually repre- 
sents the function or not. Kiemann rejected Gauchy's defini- 
tion of a definite integral on account of its arbitrariness, gave 
a new definition, and then inquired when a function has an 
integral. His researches brought to light the fact that con- 
tinuous functions need not always have a differential coeffi- 
cient. But this property, which was shown by Weierstrass to 
belong to large classes of functions, was not found necessarily 
to exclude them from being represented by Fourier's series. 
Doubts on some of the conclusions about Fourier's series were 
thrown by the observation, made by Weierstrass, that the 
integral of an infinite series can be shown to be equal to the 
sum of the integrals of the separate terms only when the series 
converges uniformly within the region in question. The sub- 
ject of uniform convergence was investigated by Philipp Lud- 
wig Seidel (1848) and G. G. Stokes (1847), and has assumed 
great importance in Weierstrass' theory of functions. It 
became necessary to prove that a trigonometric series repre- 
senting a continuous function converges uniformly. This was 
done by Heinrich Eduard Heine (1821-1881), of Halle. Later 
researches on Fourier's series were made by G. Cantor and 
Du Bois-Reymond. 

As compared with the vast development of other mathe- 



840 A HISTOBY OF MATHEMATICS. 

matical branches, the theory of probability has made veiy 
insignificant progress since the time of Laplace. ImproTe- 
ments and simplications in the mode of exposition have been 
made by A. De Morgan, G. Boole, A. Meyer (edited by E. 
Czuber), J. Bertrand. Goumot's and Westergaard's treatment 
of insurance and the theory of life-tables are classical Appli- 
cations of the calculus to statistics have been made by L. A. J. 
Quetelet (1796-1874), director of the observatory at Brussels; 
by Lexis ; Harald Westergaard, of Copenhagen ; and Diising. 

Worthy of note is the rejection of inverse probability by the 
best authorities of our time. This branch of probability had 
been worked out by Thomas Bayes (died 1761) and by Laplace 
(Bk. II., Ch. VI. of his Th^orie Analytiqae). By it some 
logicians have explained induction. For example, if a man, 
who has never heard of the tides, were to go to the shore of the 
Atlantic Ocean and witness on m successive days the rise of the 
sea, then, says Quetelet, he would be entitled to conclude that 

there was a probability equal to that the sea would rise 

next day. Putting m = 0, it is seen that this view rests upon 
the unwarrantable assumption that the probability of a totally 
unknown event is \, or that of all theories proposed for inves- 
tigation one-half are true. W. S. Jevons in his Principles of 
Science founds induction upon the theory of inverse proba- 
bility, and F. Y. Edgeworth also accepts it in his Mathematical 
Psychics, 

The only noteworthy recent addition to probability is the 
subject of " local probability," developed by several English 
and a few American and French mathematicians. The earliest 
problem on this subject dates back to the time of Buffon, the 
naturalist, who proposed the problem, solved by himself and 
Laplace, to determine the probability that a short needle, 
thrown at random upon a floor ruled with equidistant parallel 



ANALYSIS. 841 

lineSy will fall on one of the lines. Then came Sylvester's 
four-point problem: to find the probability that four points, 
taken at random within a given boundary, shall form a re- 
entrant quadrilateral. Local probability has been studied in 
England by A. R. Clarke, H. McCoU, S. Watson, J. Wolsten- 
holme, but with greatest success by M, W, Crofton of the 
military school at Woolwich. It was pursued in America by 
E. B. Seitz ; in France by C. Jordan, E. Lemoine, E. Barbier, 
and others. Through considerations of local probability, 
Crofton was led to the evaluation of certain definite integrals. 
The first full scientific treatment of differential equations 
was given by Lagrange and Laplace. This remark is especially 
true of partial differential equations. The latter were investi- 
gated in more recent time by Monge, Pfaff, Jacobi, fimile Bour 
(1831-1866) of Paris, A. Weiler, Clebsch, A. N. Korkine of St. 
Petersburg, G. Boole, A. Meyer, Cauchy, Serret, Sophus Lie, 
and others. In 1873 their reseaches, on partial differential 
equations of the first order, were presented in text-book form 
by Paul Mansion, of the University of Gand. The keen 
researches of Johann Friedrich Pfaff (1795-1825) marked a 
decided advance. He was an intimate friend of young Gauss 
at Gottingen. Afterwards he was with the astronomer Bode. 
Later he became professor at Helmstadt, then at Halle. By a 
peculiar method, Pfaff found the general integration of par- 
tial differential equations of the first order for any number 
of variables. Starting from the theory of ordinary differential 
equations of the first order in n variables, he gives first their 
general integration, and then considers the integration of the 
partial differential equations as a particular case of the former, 
assuming, however, as known, the general integration of differ- 
ential equations of any order between two variables. His 
researches led Jacobi to introduce the name ^'Pfafi&an prob- 
lem." From the connection, observed by Hamilton, between 



342 A HISTORY OP MATHEMATICS. 

a system of ordinary differential equations (in analjrtical 
mechanics) and a partial differential equation, Jacobi drew 
the conclusion that, of the series of systems whose successive 
integralTion Pfaff's method demanded, all but the first system 
were entirely superfluous. Clebsch considered Pfaff's problem 
from a new point of view, and reduced it to systems of simul- 
taneous linear partial differential equations, which can be 
established independently of each other without any integra- 
tion. Jacobi materially advanced the theory of differential 
equations of the first order. The problem to determine un- 
known functions in such a way that an integral containing 
these functions and their differential coefficients, in a pre- 
scribed manner, shall reach a maximum or minimum value, 
demands, in the first place, the vanishing of the first variation 
of the integral. This condition leads to differential equations^ 
the integration of which determines the functions. To ascer- 
tain whether the value is a maximum or a minimum, the 
second variation must be examined. This leads to new and 
difficult differential equations, the integration of which, for 
the simpler cases, was ingeniously deduced by Jacobi from 
the integration of the differential equations of the first varia- 
tion. Jacobins solution was perfected by Hesse, while Clebsch 
extended to the general case Jacobi's results on the second 
variation. Cauchy gave a method of solving partial differ- 
ential equations of the first order having any number of 
variables, which was corrected and extended by Serret, J. Ber- 
trand, 0. Bonnet in France, and Imschenetzky in Kussia. 
Fundamental is the proposition of Cauchy that every ordinary 
differential equation admits in the vicinity of any non-singular 
point of an integral, which is synectic within a certain circle 
of convergence, and is developable by Taylor's theorem. 
Allied to the point of view indicated by this theorem is that 
of Kiemann, who regards a function of a single variable as 



ANALYSIS. 848 

defined by the position and nature of its singularities, and 
who has applied this conception to that linear differential 
equation of the second order, which is satisfied by the hyper- 
geometric series. This equation was studied also by Gauss 
and Kummer. Its general theory, when no restriction is 
imposed upon the value of the variable, has been considered 
by J. Tannery, of Paris, who employed Fuchs' method of 
linear differential equations and found all of Rummer's 
twenty-four integrals of this equation. This study has been 
continued by fidouard Groursat of Paris. 

A standard text-book on Differential Equations, including 
original matter on integrating factors, singular solutions, and 
especially on symbolical methods, was prepared in 1859 by 
George Boole (1815-1864), at one time professor in Queen's 
University, Cork, Ireland. He was a native of Lincoln, and a 
self-educated mathematician of great power. His treatise on 
Finite Differences (1860) and bis Laws of Thought (1854) are 
works of high merit. 

The fertility of the conceptions of Cauchy and Biemann 
with regard to differential equations is attested by the 
researches to which they have given rise on the part of 
Laxarus Fachs of Berlin (born 1885), Felix Klein of Gottingen 
(bom 1849), Henri Poincare of Paris (bom 1854), and others. 
The study of linear differential equations entered a new 
period with the publication of Fuchs' memoirs of 1866 
and 1868. Before this, linear equations with constant co- 
efficients were almost the only ones for which general methods 
of integration were known. While the general theory of 
these equations has recently been presented in a new light 
by Hermite, Darboux, and Jordan, Fuchs began the study 
from the more general standpoint of the linear differential 
equations whose coefficients are not constant. He directed 
his attention mainly to those whose integrals are all regular. 



844 A HISTOKY OF MATHEMATICS. 

If the variable be made to describe all possible paths enclos- 
ing one or more of the critical points of the equation, we 
have a certain substitution corresponding to each of the 
paths; the aggregate of all these substitutions being called 
a group. The forms of integrals of such equations were 
examined by Fuchs and by G. Frobenius by independent 
methods. Logarithms generally appear in the integrals of 
a group, and Fuchs and Frobenius investigated the conditions 
under which no logarithms shall appear. Through the stady 
of groups the reducibility or irreducibility of linear differ- 
ential equations has been examined by Frobenius and Leo 
Konigsberger. The subject of linear differential equations, 
not all of whose integrals are regular, has been attacked by 
G. Frobenius of Berlin, W. Thom^ of Greifswald (bom 1841) , 
and Poincar6, but the resulting theory of irregular integrals 
is as yet in very incomplete form. 

The theory of invariants associated with linear differential 
equations has been developed by Halphen and by A. R. Forsyth. 

The researches above referred to are closely connected with 
the theory of functions and of groups. Endeavours have thus 
been made to determine the nature of the function defined by 
a differential equation from the differential equation itself, 
and not from any analytical expression of the function, obtained 
first by solving the differential equation. Instead of studying 
the properties of the integrals of a differential equation for all 
the values of the variable, investigators at first contented them- 
selves with the study of the properties in the vicinity of a 
given point. The nature of the integrals at singular points 
and at ordinary points is entirely different. Albert Briot 
(1817-1882) and Jean Claude Bouquet (1819-1885), both of 
Paris, studied the case when, near a singular point, the dif- 
ferential equations take the form {x — x^ -^ = I {xy). Fuchs 

dx J 



ANALYSIS. 845 

gave the development in series of the integrals for the partic- 
ular case of linear equations. Poincar6 did the same for the 
case when the equations are not linear, as also for partial 
differential equations of the first order. The developments for 
ordinary points were given by Cauchy and Madame Kowalevsky. 
The attempt to express the integrals by developments that 
are always convergent and not limited to particular points in 
a plane necessitates the introduction of new transcendents, for 
the old functions permit the integration of only a small num- 
ber of differential equations. Poincar^ tried this plan with 
linear equations, which were then the best known, having 
been studied in the vicinity of given points by Fuchs, Thom6, 
Frobenius, Schwarz, Klein, and Halphen. Confining himself to 
those with rational algebraical coefficients, Poincar^ was able 
to integrate them by the use of functions named by him Fuchr 
sians,^ He divided these equations into " families." If the 
integral of such an equation be subjected to a certain trans- 
formation, the result will be the integral of an equation 
belonging to the same family. The new transcendents have a 
great analogy to elliptic functions; while the region of the 
latter may be divided into parallelograms, each representing a 
group, the former may be divided into curvilinear polygons, 
so that the knowledge of the function inside of one polygon 
carries with it the knowledge of it inside the others. Thus 
Poincar^ arrives at what he calls FucJman groups. He found, 
moreover, that Fuchsian functions can be expressed as the 
ratio of two transcendents (theta-fuchsians) in the same way 
that elliptic functions can be. If, instead of linear substitu- 
tions with real coefficients, as employed in the above groups, 
imaginary coefficients be used, then discontinuous gi'oups are 
obtained, which he called Kleinians. The extension to non- 
linear equations of the method thus applied to linear equar 
tions has been begun by Fuchs and Poincar^. 



346 A HISTORY OF MATHEMATICS. 

We have seen that among the earliest of the several kinds 
of '< groups" are the finite discontinuous groups (groups in 
the theory of substitution), which since the time of Galois 
have become the leading concept in the theory of algebraic 
equations; that since 1876 Felix Klein, H. Poincar^ and 
others have applied the theory of finite and infinite discontin- 
uous groups to the theory of functions and of differential 
equations. The finite continuous groups were first made 
the subject of general research in 1873 by Sophus Lie, now 
of Leipzig, and applied by him to the integration of ordinary 
linear partial differential equations. 

Much interest attaches to the determination of those linear 
differential equations which can be integrated by simpler 
functions, such as algebraic, elliptic, or Abelian. This has 
been studied by C. Jordan, P. Appel of Paris (bom 1858), and 
Poincar^. 

The mode of integration above referred to, which makes 
known the properties of equations from the standpoint of the 
theory of functions, does not suffice in the application of 
differential equations to questions of mechanics. If we con- 
sider the function as defining a plane curve, then the general 
form of the curve does not appear from the above mode of 
investigation. It is, however, often desirable to construct 
the curves defined by differential equations. Studies having 
this end in view have been carried on by Briot and Bouquet, 
and by Poincar^.®^ 

The subject of singular solutions of differential equations 
has been materially advanced since the time of Boole by G. 
Darboux and Cayley. The papers prepared by these mathe- 
maticians point out a difficulty as yet unsurmounted : whereas 
a singular solution, from the point of view of the integrated 
equation, ought to be a phenomenon of universal, or at least of 
general occurrence, it is, on the other hand, a very special and 



THEORY OF FUNCTIONS. 847 

exceptional phenomenon from the point of view of the differ- 
ential equation.® A geometrical theory of singular solutions 
resembling the one used by Cayley was previously employed 
by W. W. Johnson of Annapolis. 

An advanced Treatise on Linear Differential Equations 
(1889) was brought out by Thomas Craig of the Johns Hop- 
kins University. He chose the algebraic method of presenta- 
tion followed by Hermite and Poincar^^ instead of the geometric 
method preferred by Klein and Schwarz. A notable work, the 
Traiti d^AnaXyse, is now being published by ifimile Picard of 
Paris, the interest of which is made to centre in the subject of 
differential equations. 



THEORY OF FUNCTIONS. 

We begin our sketch of the vast progress in the theory of 
functions by considering the special class called elliptic func- 
tions. These were richly developed by Abel and Jacobi. 

Kiels Henrick Abel (1802-1829) was born at Findoe in Nor- 
way, and was prepared for the university at the cathedral 
school in Christiania. He exhibited no interest in mathe- 
matics until 1818, when B. Holmboe became lecturer there, 
and aroused Abel's interest by assigning original problems 
to the class. Like Jacobi and many other young men who 
became eminent mathematicians, Abel found the first exercise 
of his talent in the attempt to solve by algebra the general 
equation of the fifth degree. In 1821 he entered the Uni- 
versity in Christiania. The works of Euler, Lagrange, and 
Legendre were closely studied by him. The idea of the inver- 
sion of elliptic functions dates back to this time. His extraor- 
dinary success in mathematical study led to the offer of a 
stipend by the government, that he might continue his studies 



348 A HISTORY OF MATHEMATICS. 

in Grermany and France. Leaving Norway in 1825, Abel vis- 
ited the astronomer, Schumacher, in Hamburg, and spent six 
months in Berlin, where he became intimate with Aognst 
Leopold Crelle (1780-1855), and met Steiner. Encouraged by 
Abel and Steiner, Crelle started his journal in 1826. Abel 
began to put some of his work in shape for print. His proof 
of the impossibility of solving the general equation of the fifth 
degree by radicals, — first printed in 1824 in a very concise 
form, and difficult of apprehension, — was elaborated in greater 
detail, and published in the first volume. He entered also 
upon the subject of infinite series (particularly the binomial 
theorem, of which he gave in CreUe^s Journal a rigid general 
investigation), the study of functions, and of the integral 
calculus. The obscurities everywhere encountered by him 
owing to the prevailing loose methods of analysis he endeav- 
oured to clear up. For a short time he left Berlin for Frei- 
berg, where he had fewer interruptions to work, and it was 
there that he made researches on hyperelliptic and Abelian 
functions. In July, 1826, Abel left Germany for Paris with- 
out having met Gauss ! Abel had sent to Gauss his proof of 
1824 of the impossibility of solving equations of the fifth 
degree, to which Gauss never paid any attention. This slight, 
and a haughtiness of spirit which he associated with Gauss, 
prevented the genial Abel from going to Gottiugen. A similar 
feeling was entertained by him later against Cauchy. Abel 
remained ten months in Paris. He met there Dirichlet, 
Legendre, Cauchy, and others; but was little appreciated. 
He had already published several important memoirs in 
CrelWs JowyicU, but by the French this new periodical was 
as yet hardly known to exist, and Abel was too modest to 
speak of his own work. Pecuniary embarrassments induced 
him to return home after a second short stay in Berlin. At 
Christiania he for some time gave private lessons, and served 



THBORY OP FUNCTIONS. 849 

as docent. Crelle secured at last an appointment for him at 
Berlin ; but the news of it did not reach Norway until after 
the death of Abel at Froland." 

At nearly the same time with Abel, Jacobi published articles 
on elliptic functions. Legendre's favourite subject, so long 
neglected, was at last to be enriched by some extraordinary 
discoveries. The advantage to be derived by inverting the 
elliptic integral of the first kind and treating it as a function 
of its amplitude (now called elliptic function) was recognised 
by Abel, and a few months later also by Jacobi. A second 
fruitful idea, also arrived at independently by both, is the 
introduction of imaginaries leading to the observation that 
the new functions simulated at once trigonometric and expo- 
nential functions. For it was shown that while trigonometric 
functions had only a real period, and exponential only an imag- 
inary, elliptic functions had both sorts of periods. These two 
discoveries were the foundations upon which Abel and Jacobi, 
each in his own way, erected beautiful new structures. Abel 
developed the curious expressions representing elliptic func- 
tions by infinite series or quotients of infinite products. 
Great as were the achievements of Abel in elliptic functions, 
they were eclipsed by his researches on what are now called 
Abelian functions. Abel's theorem on these functions was 
given by him in several forms, the most general of these 
being that in his Mhnoire sur une propriiti ght^rcde d'une 
dasse tr^8-4tendtie de fonctions transcendentes (1826). The his- 
tory of this memoir is interesting. A few months after his 
arrival in Paris, Abel submitted it to the French Academy. 
Gauchy and Legendre were appointed to examine it ; but said 
nothing about it until after Abel's death. In a brief statement 
of the discoveries in question, published by Abel in CreUU^s 
Journal, 1829, reference is made to that memoir. This led 
Jacobi to inquire of Legendre wha% had become *of it. Le- 



350 A HISTOBY OF MATHEMATICS. 

gendre says that the manuscript was so badly written as to be 
illegible, and that Abel was asked to hand in a better copy, 
which he neglected to do. The memoir remained in Canchy's 
hands. It was not published until 1841. By a singular mis- 
hap, the manuscript was lost before the proof-sheets were 
read. 

In its form, the contents of the memoir belongs to the inte- 
gral calculus. Abelian integrals depend upon an irrational 
function y which is connected with x by an algebraic equa- 
tion F{Xj y) = 0. Abel's theorem asserts that a sum of such 
integrals can be expressed by a definite number p of similar 
integrals, where p depends merely on the properties of the 
equation ^(a;, ^) = 0. It was shown later that p is the defi- 
ciency of the curve F{Xy y) = 0. The addition theorems of 
elliptic integrals are deducible from Abel's theorem. The 
hyperelliptic integrals introduced by Abel, and proved by him 
to possess multiple periodicity, are special cases of Abelian 
integrals whenever p= or > 3. The reduction of Abelian to 
elliptic integrals has been studied mainly by Jacobi, Her mite, 
Konigsberger, Brioschi, Goursat, E. Picard, and 0. Bolza of 
the University of Chicago. 

Two editions of Abel's works have been published : the first 
by Holmboe in 1839, and the second by Sylow and Lie in 
1881. 

Abel's theorem was pronounced by Jacobi the greatest dis- 
covery of our century on the integral calculus. The aged 
Legendre, who greatly admired Abel's genius, called it "moTi- 
umentum aere perenniusy During the few years of work 
allotted to the young Norwegian, he penetrated new fields of 
research, the development of which has kept mathematicians 
busy for over half a century. 

Some of the discoveries of Abel and Jacobi were anticipated 
by Gauss. In the Disquisithnes Arithmeticce he observed 



THEORY OF FUNCTIONS. 861 

that the principles which he used in the division of the circle 
were applicable to many other functions, besides the circular, 
and particularly to the transcendents dependent on the integral 

-y===' From this Jacobi" concluded that Oauss had 

thirty years earlier considered the nature and properties of 
elliptic functions and had discovered their double periodicity. 
The papers in the collected works of Grauss confirm this con- 
clusion. 

Carl Oustav Jacob Jacobi^ (1804-1851) was born of Jewish 
parents at Potsdam. Like many other mathematicians he was 
initiated into mathematics by reading Euler. At the Univer- 
sity of Berlin, where he pursued his mathematical studies 
independently of the lecture courses, he took the degree of 
Ph.D. in 1825. After giving lectures in Berlin for two years, 
he was elected extraordinary professor at Konigsberg, and two 
years later to the ordinary professorship there. After the 
publication of his Fundamenta Nova he spent some time in 
travel, meeting Gauss in Gottingen, and Legendre, Fourier, 
Poisson, in Paris. In 1842 he and his colleague, Bessel, at- 
tended the meetings of the British Association, where they 
made the acquaintance of English mathematicians. 

His early researches were on Gauss' approximation to the 
value of definite integrals, partial differential equations, Le- 
gendre's coefficients, and cubic residues. He read Legendre's 
Exerdsesj which give an account of elliptic integrals. When 
he returned the book to the library, he was depressed in spirits 
and said that important books generally excited in him new 
ideas, but that this time he had not been led to a single origi- 
nal thought. Though slow at first, his ideas flowed all the 
richer afterwards. Many of his discoveries in elliptic func- 
tions were made independently by Abel. Jacobi communicated 
his first researches to CreU^s Journal. In 1829, at the age 



352 A HISTORY OF MATHEMATIGS. 

of twenty-five, he published his Fundamenta Nova 
Functionum EUipticarum, which contains in condensed form 
the main results in elliptic functions. This work at once 
secured for him a wide reputation. He then made a closer 
study of thetarfunctions and lectured to his pupils on a new 
theory of elliptic functions based on the theta-functions. He 
developed a theory of transformation which led him to a miQ- 
titude of formulae containing q, a transcendental function of 
the modulus, defined by the equation q =e~''*'^. He was also 
led by it to consider the two new functions H and 0, which 
taken each separately with two different arguments are the 
four (single) theta-functions designated by the ^®^%^ 04.'' 
In a short but very important memoir of 1832, he shows that 
for the hyperelliptic integral of any class the direct functions 
to which Abel's theorem has reference are not functions of a 
single variable, such as the elliptic an, en, dn, but functions of 
p variables." Thus in the case 2? = 2, which Jacobi especially 
considers, it is shown that Abel's theorem has reference to 
two functions X(w, v), Xi(u, v), each of two variables, and 
gives in effect an addition-theorem for the expression of the 
functions \{u -\-u\ v -f v'), Xi(w -f u\ v -f v') algebraically in 
terms of the functions \{u,v), Xi(w, v), \(u',v'), \i(u'fV'), By 
the memoirs of Abel and Jacobi it may be considered that the 
notion of the Abelian function of p variables was established 
and the addition-theorem for these functions given. Recent 
studies touching Abelian functions have been made by Weier- 
strass, E. Picard, Madame Kowalevski, and Poincar^. Jacobi's 
work on differential equations, determinants, dynamics, and 
the theory of numbers is mentioned elsewhere. 

In 1842 Jacobi visited Italy for a few months to recuperate 
his health. At this time the Prussian government gave him 
a pension, and he moved to Berlin, where the last years of his 
life were spent. 



THEOBY OF FUNCTIONS. 868 

The researches on functions mentioned thus far have been 
greatly extended. In 1858 Charles Hermite of Paris (bom 1822), 
introduced in place of the variable q of Jacobi a new variable « 
connected with it by the equation q = e'**, so that a> = ffe'/A:, and 
was led to consider the functions ^(w), ^(w), x(«) •" Henry 
Smith regarded a theta-function with the argument equal to 
zero, as a function of <i>. This he called an omega-function, 
while the three functions ^(<i>), ^(«i>)> x(^)> ^® ^^ modular 
functions. Eesearches on theta-functions with respect to real 
and imaginary arguments have been made by Meissel of Kiel, 
J. Thomae of Jena, Alfred Enneper of Gottingen (1830-1886). 
A general formula for the product of two theta-functions was 
given in 1854 by H. Schroter of Breslau (1829-1892). These 
functions have been studied also by Cauchy, Konigsberger of 
Heidelberg (born 1837), F. S. Richelot of Konigsberg (1808- 
1875), Johann (Jeorg Rosenham of Konigsberg (1816-1887), 
L. Schliifli of Bern (bom 1818). « 

Legendre's method of reducing an elliptic differential to its 
normal form has called forth many investigations, most impor- 
tant of which are those of Eichelot and of Weierstrass of 
Berlin. 

The algebraic transformations of elliptic functions involve 
a relation between the old modulus and the new one which 
Jacobi expressed by a differential equation of the third order, 
and also by an algebraic equation, called by him ''modular 
equation." The notion of modular equations was familiar to 
Abel, but the development of this subject devolved upon later 
investigators. These equations have become of importance in 
the theory of algebraic equations, and have been studied by 
Sohnke, E. Mathieu, L. Konigsberger, E. Betti of Pisa (died 
1892), C. Hermite of Paris, Joubert of Angers, Francesco 
Brioschi of Milan. Schl&fli, H. Schr5ter, M. Oudermann of 
Cleve, G«tzlaff. 



854 A HI8T0BY OF MATHEMATICS. 

Felix Klein of Grottingen has made an extensive study of 
modular functions, dealing with a type of operations lying 
between the two extreme types, known as the theory of substi- 
tutions and the theory of invariants and covariants. Klein's 
theory has been presented in book-form by his pupil, Robert 
Fricke. The bolder features of it were first published in his 
Ikosaeder, 1884. His researches embrace the theory of mod- 
ular functions as a specific class of elliptic functions, the 
statement of a more general problem as based on the doctrine 
of groups of operations, and the further development of the 
subject in connection with a class of Kiemann's surfaces. 

The elliptic functions were expressed by Abel as quotients 
of doubly infinite products. He did not, however, inquire 
rigorously into the convergency of the products. In 1845 
Cayley studied these products, and found for them a complete 
theory, based in part upon geometrical interpretation, which 
he made the basis of the whole theory of elliptic functions. 
Eisenstein discussed by purely analytical methods the general 
doubly infinite product, and arrived at results which have 
been greatly simplified in form by the theory of primary 
factors, due to Weierstrass. A certain function involving a 
doubly infinite product has been called by Weierstrass the 
sigma-function, and is the basis of his beautiful theory of 
elliptic functions. The first systematic presentation of Weier- 
strass' theory of elliptic functions was published in 1886 by 
G. H. Halphen in his Th4orie des fonctions elliptiquea et des 
leurs applications. Applications of these functions have been 
given also by A. G. Greenhill. Generalisations analogous to 
those of Weierstrass on elliptic functions have been made 
by Felix Klein on hyperelliptic functions. 

Standard works on elliptic functions have been published by 
Briot and Bouquet (1859), by Konigsberger^ Cayley ^ Heinrich 
Dur^e of Prague (1821-1893), and others. 



THEORY OF FUNCTIONS. 866 

Jacobi's work on Abelian and thetarfunctions was greatly 
extended by Adolph Gopel (1812-1847), professor in a gym- 
nasium near Potsdam, and Johann Georg Rosenhain of Konigs- 
berg (1816-1887). Gopel in his TheoricE transcendentium primi 
ordinis adumbratio levis (Crelle, 35, 1847) and Kosenhain in 
several memoirs established each independently, on the analogy 
of the single theta-functions, the functions of two variables, 
called double theta-functions, and worked out in connection 
with them the theory of the Abelian functions of two variables. 
The theta-relations established by Gopel and Kosenhain re- 
ceived for thirty years no further development, notwithstand- 
ing the fact that the double theta series came to be of increasing 
importance in analytical, geometrical, and mechanical prob- 
lems, and that Hermite and Konigsberger had considered the 
subject of transformation. Finally, the investigations of G. W. 
Borchardt of Berlin (1817-1880), treating of the representation 
of Rummer's surface by Gopel's biquadratic relation between 
four theta-functions of two variables, and researches of H. H. 
Weber of Marburg, F. Prym of Wiirzburg, Adolf Krazer, and 
Martin Krause of Dresden led to broader views. Besearches 
on double theta-functions, made by Cayley, were extended to 
quadruple theta-functions by Thomas Graig of the Johns 
Hopkins University. 

Starting with the integrals of the most general form and 
considering the inverse functions corresponding to these in- 
tegrals (the Abelian functions of p variables), Biemann 
defined the theta-functions of p variables as the sum of a 
j>-tuply infinite series of exponentials, the general term de- 
pending on p variables. Biemann shows that the Abelian 
functions are algebraically connected with thetarf unctions of 
the proper arguments, and presents the theory in the broadest 
form.** He rests the theory of the multiple thetarfunctions 
upon the general principles of the theory of functions of a 
complex variable. 



856 A HISTORY OF MATHEMATICS. 

Through the researches of A. Brill of Ttibingex^ M. Ndther 
of Erlangen, and Ferdinand Lindemann of Munich, made 
in connection with Eiemann-Koch's theorem and the theory 
of residuation, there has grown out of the theory of Abelian 
functions a theory of algebraic functions and point-groups on 
algebraic curves. 

Before proceeding to the general theory of functions, we 
make mention of the " calculus of functions," studied chiefly 
by C. Babbage, J. F. W. Herschel, and De Morgan, which was 
not so much a theory of functions as a theory of the solution 
of functional equations by means of known functions or 
symbols. 

The history of the general theory of functions begins with 
the adoption of new definitions of a function. With the 
BernouUis and Leibniz, y was called a function of x, if there 
existed an equation between these variables which made it 
possible to calculate y for any given value of x lying any- 
where between — oo and + oo. The study of Fourier's theory 
of heat led Dirichlet to a new definition : y is called a function 
of flj, if y possess one or more definite values for each of certain 
values that x is assumed to take in an interval a^ to «i. In 
functions thus defined, there need be no analytical connection 
between y and aj, and it becomes necessary to look for possible 
discontinuities. A great revolution in the ideas of a function 
was brought about by Cauchy when, in a fimction as defined 
by Dirichlet, he gave the variables imaginary values, and when 
he extended the notion of a definite integral by letting the 
variable pass from one limit to the other by a succession of 
imaginary values along arbitrary paths. Cauchy established 
several fundamental theorems, and gave the first great impulse 
to the study of the general theory of functions. His researches 
were continued in France by Puiseux and Liouville. But more 
profound investigations were made in Germany by Eiemann. 



THEORY OF FUNCTIONS. 867 

Oeorg Friedrich Bemhard Riemann (1826-1866) was born at 
Breselenz in Hanover. His father wished him to stndy 
theology, and he accordingly entered upon philological and 
theological studies at Gottingen. He attended also some lec- 
tures on mathematics. Such was his predilection for this 
science that he abandoned theology. After studying for a 
time under Gauss and Stem, he was drawn, in 1847, to Berlin 
by a galaxy of mathematicians, in which shone Dirichlet, 
Jacobi, Steiner, and Eisenstein. Returning to Gottingen in 
1850, he studied physics under Weber, and obtained the 
doctorate the following year. The thesis presented on that 
occasion, OruncUagen fur eine allgemeine Theorie der Funktionen 
einer ver&nderlichen complexen Chrbsse, excited the admiration of 
Gauss to a very unusual degree, as did also Kiemann's trial 
lecture, Uebw die Hypotheaen welche der Qeom^rie zu Orunde 
liegen, Kiemann's Habilitationsschrift was on the Bepresen- 
tation of a Function by means of a Trigonometric Series, in 
which he advanced materially beyond the position of Dirich- 
let. Our hearts are drawn to this extraordinarily gifted but 
shy genius when we read of the timidity and nervousness 
displayed when he began to lecture at Gdttingen, and of his 
jubilation over the unexpectedly large audience of eight students 
at his first lecture on differential equations. 

Later he lectured on Abelian functions to a class of three 
only, — Schering, Bjerknes, and Dedekind. Gauss died in 1855, 
and was succeeded by Dirichlet. On the death of the latter, 
in 1859, Eiemann was made ordinary professor. In 1860 he 
visited Paris, where he made the acquaintance of French 
mathematicians. The delicate state of his health induced 
him to go to Italy three times. He died on his last trip at 
Selasca, and was buried at Biganzolo. 

Like all of Riemann's researches, those on functions were 
profound and far-reaching. He laid the foundation for a 



858 A HISTORY OF MATHEMATICS. 

general theory of functions of a complex variable. The theory 
of potential, which up to that time had been used only in 
mathematical physics, was applied by him in pure mathe- 
matics. He accordingly based his theory of functions on the 

partial differential equation, t-s + -^-s = ^w = 0, which must 

hold for the analytical function w^u-^-iv of zs=x+iy. It 
had been proved by Dirichlet that (for a plane) there is always 
one, and only one, function of x and y, which satisfies Au = 0, 
and which, together with its differential quotients of the first 
two orders, is for all values of x and y within a given area 
one-valued and continuous, and which has for points on the 
boundary of the area arbitrarily given values.* Kiemann 
called this " Dirichlet's principle," but the same theorem was 
stated by Green and proved analytically by Sir William 
Thomson. It follows then that w is uniquely determined for 
all points within a closed surface, if u is arbitrarily given 
for all points on the curve, whilst v is given for one point 
within the curve. In order to treat the more complicated 
case where w has n values for one value of 2, and to observe 
the conditions about continuity, Riemann invented the cele- 
brated surfaces, known as "Riemann's surfaces," consisting 
of n coincident planes or sheets, such that the passage from 
one sheet to another is made at the branch-points, and that the 
n sheets form together a multiply-connected surface, which 
can be dissected by cross-cuts into a singly-connected surface. 
The n-valued function w becomes thus a one-valued function. 
Aided by researches of J. Liiroth of Freiburg and of Clebsch, 
W. K. Clifford brought Riemann's surface for algebraic functions 
to a canonical form, in which only the two last of the n leaves 
are multiply-connected, and then transformed the surface into 
the surface of a solid with p holes. A. Hurwitz of Ziirich 
discussed the question, how far a Riemann's surface is deter- 



THEORY OP FUNCTIONS. 859 

minate by the assignment of its number of sheets, its branch- 
points and branch-lines." 

Biemann's theory ascertains the criteria which will deter- 
mine an analytical function by aid of its discontinuities and 
boundary conditions, and thus deiines a function indepen- 
dently of a mathematical expression. In order to show that 
two different expressions are identical, it is not necessary to 
transform one into the other, but it is sufficient to prove the 
agreement to a far less extent, merely in certain critical points. 

Biemann's theory, as based on Dirichlet's principle (Thom- 
son's theorem), is not free from objections. It has become 
evident that the existence of a derived function is not a con- 
sequence of continuity, and that a function may be integrable 
without being differentiable. It is not known how far the 
methods of the infinitesimal calculus and the calculus of 
variations (by which Dirichlet's principle is established) can 
be applied to an unknown analytical function in its generality. 
Hence the use of these methods will endow the functions with 
properties which themselves require proof. Objections of this 
kind to Biemann's theory have been raised by Kronecker, 
Weierstrass, and others, and it has become doubtful whether 
his most important theorems are actually proved. In con- 
sequence of this, attempts have been made to graft Biemann's 
speculations on the more strongly rooted methods of Weier- 
strass. The latter developed a theory of functions by start- 
ing, not with the theory of potential, but with analytical 
expressions and operations. Both applied their theories to 
Abelian functions, but there Biemann's work is more gen- 
eral.* 

The theory of functions of one complex variable has been 
studied since Biemann's time mainly by Karl WeierstraM of 
Berlin (born 1815), Outtaf Bfittag-Leffler of Stockholm (bom 
1846), and Poincar^ of Paris. Of the three classes of such 



860 A HISTOBY OF MATHEMATIOS. 

functions (viz. functions uniform throughout, functions uni- 
form only in lacunary spaces, and non-uniform fonctions) 
Weierstrass showed that those functions of the first class 
which can be developed according to ascending powers of x 
into converging series, can be decomposed into a product of an 
infinite number of primary factors. A primary factor of the 

species n is the product (1 — - je''(*), P(.) being an entire poly- 
nomial of the nth degree. A function of the species n is one, 
all the primary factors of which are of species n. This classi- 
fication gave rise to many interesting problems studied also 
by Poincar^. 

The first of the three classes of functions of a complex 
variable embraces, among others, functions having an infinite 
number of singular points, but no singular lines, and at the 
same time no isolated singular points. These are Fuchsian 
functions, existing throughout the whole extent. Poincar^ 
first gave an example of such a function. 

Uniform functions of two variables, unaltered by certain 
linear substitutions, called hyperfuchsian functions, have been 
studied by E. Picard of Paris, and by Poincard*^ 

Functions of the second class, uniform only in lacunary 
spaces, were first pointed out by Weierstrass. The Fuchsian 
and the Kleinian functions do not generally exist, except in 
the interior of a circle or of a domain otherwise bounded, and 
are therefore examples of functions of the second class. 
Poincar^ has shown how to generate functions of this class, 
and has studied them along the lines marked out by Weier- 
strass. Important is his proof that there is no way of 
gfeneralising them so as to get rid of the lacunae. 

Non-uniform functions are much less developed than the 
preceding classes, even though their properties in the vicinity 
of a given point have been diligently studied, and though 



THBORY OF FUNCTIONS. 361 

much light has been thrown on them by the use of Biemann's 
surfaces. With the view of reducing their study to that of 
uniform transcendents, Poincard proved that if y is any 
analytical non-uniform function of x, one can always find a 
yariable Zy such that x and y are uniform functions of z, 

Weierstrass and Darboux have each given examples of con- 
tinuous functions having no derivatives. Formerly it had 
been generally assumed that every function had a derivative. 
Ampere was the first who attempted to prove analytically 
(1806) the existence of a derivative, but the demonstration 
is not valid. In treating of discontinuous functions, Darboux 
established rigorously the necessary and sufficient condition 
that a continuous or discontinuous function be susceptible of 
integration. He gave fresh evidence of the care that must 
be exercised in the use of series by giving an example of a 
series always convergent and continuous, such that the series 
formed by the integrals of the terms is always convergent, and 
yet does not represent the integral of the first series.^ 

The general theory of functions of two variables has been 
investigated to some extent by Weierstrass and Poincar^. 

H. A. Schwarz of Berlin (born 1845), a pupil of Weierstrass, 
has given the conform representation {Abbildung) of various 
surfaces on a circle. In transforming by aid of certain 
substitutions a polygon bounded by circular arcs into another 
also bounded by circular arcs, he was led to a remarkable 
differential equation ^ (u', £)=:^(u, t), where ^ (u, t) is the 
expression which Cayley calls the ^'Schwarzian derivative," 
and which led Sylvester to the theory of reciprocants. 
Schwarz's developments on minimum surfaces, his work on 
hypergeometric series, his inquiries on the existence of solu- 
tions to important partial differential equations under prescribed 
conditions, have secured a prominent place in mathematical 
literature. 



362 A HISTORY OF MATHEMATICS. 

The modem theory of functions of one real variable was 
first worked out by H. Kankel, Dedekind, Q. Cantor, Dini, and 
Heine, and then carried further, principally, by Weierstrass, 
Schwarz, Du Bois-Keymond, Thomae, and Darboux. Hankel 
established the principle of the condensation of singularities ; 
Dedekind and Cantor gave definitions for irrational numbers; 
definite integrals were studied by Thomae, Du Bois-Keymond, 
and Darboux along the lines indicated by the definitions of 
such integrals given by Cauchy, Dirichlet, and Kiemann. Dini 
wrote a text-book on functions of a real variable (1878), which 
was translated into German, with additions, by J. Liiroth and 
A. Schepp. Important works on the theory of functions are 
the Cours de M. Hermite, Tannery's Thiorie des Fonctions 
(Tune vanable seiUe, A Treatise on the Theory of Functions by 
James Harkness and Frank Morley, and Theory of Functions of 
a Complex Variable by A, R. Forsyth, 



THEORY OF NUMBERS. 

"Mathematics, the queen of the sciences, and arithmetic, 
the queen of mathematics." Such was the dictum of Gauss, 
who was destined to revolutionise the theory of numbers. 
When asked who was the greatest mathematician in Ger- 
many, Laplace answered, Pfaff. When the questioner said 
he should have thought Gauss was, Laplace replied, "Pfaff 
is by far the greatest mathematician in Germany ; but Gauss 
is the greatest in all Europe.'^® Gauss is one of the three 
greatest masters of modern analysis, — Lagrange, Laplace, 
Gauss. Of these three contemporaries he was the youngest. 
While the first two belong to the period in mathematical his- 
tory preceding the one now under consideration, Gauss is the 
one whose writings may truly be said to mark the beginning 



THEORY OF NUMBERS. 868 

of our own epoch. In him that abundant fertility of inven- 
tion, displayed by mathematicians of the preceding period, 
is combined with an absolute rigorousness in demonstration 
which is too often wanting in their writings, and which the 
ancient Greeks might have envied. Unlike Laplace, Gauss 
strove in his writings after perfection of form. He rivals 
Lagrange in elegance, and surpasses this great Frenchman in 
rigour. Wonderful was his richness of ideas ; one thought fol- 
lowed another so quickly that he had hardly time to write 
down even the most meagre outline. At the age of twenty 
(}aus8 had overturned old theories and old methods in all 
branches of higher mathematics ; but little pains did he take 
to publish his results, and thereby to establish his priority. 
He was the first to observe rigour in the treatment of infinite 
series, the first to fully recognise and emphasise the impor- 
tance, and to make systematic use of determinants and of 
imaginaries, the first to arrive at the method of least squares, 
the first to observe the double periodicity of elliptic functions. 
He invented the heliotrope and, together with Weber, the 
bifilar magnetometer and the declination instrument. He 
reconstructed the whole of magnetic science. 

Carl Friedrich Gauss^^ (1777-1855), the son of a bricklayer, 
was born at Brunswick. He used to say, jokingly, that he 
could reckon before he could talk. The marvellous aptitude 
for calculation of the young boy attracted the attention of 
Bartels, afterwards professor of mathematics at Dorpat, who 
brought him under the notice of Charles William, Duke of 
Brunswick. The duke undertook to educate the boy, and sent 
him to the Collegium Carolinum. His progress in languages 
there was quite equal to that in mathematics. Li 1795 he 
went to Gottingen, as yet undecided whether to pursue philol- 
ogy or mathematics. Abraham Gotthelf K^tner, then pro- 
fessor of mathematics there, and now chiefly remembered for 



364 A HISTORY OF MATHEMATICS. 

his Geschichte der Mathematik (1796), was not an inspiring 
teacher. At the age of nineteen Gauss discovered a method of 
inscribing in a circle a regular polygon of seventeen sides, and 
this success encouraged him to pursue mathematics. He worked 
quite independently of his teachers, and while a student at 
Gottingen made several of his greatest discoveries. Higher 
arithmetic was his favourite study. Among his small circle 
of intimate friends was Wolfgang Bolyai. After completing 
his course he returned to Brunswick. In 1798 and 1799 he 
repaired to the university at Helmstadt to consult the library, 
and there made the acquaintance of PfafP, a mathematician of 
much power. In 1807 the Emperor of Russia offered Gauss a 
chair in the Academy at St. Petersburg, but by the advice of 
the astronomer Olbers, who desired to secure him as director 
of a proposed new observatory at Gottingen, he declined the 
offer, and accepted the place at Gottingen. Gauss had a 
marked objection to a mathematical chair, and preferred the 
post of astronomer, that he might give all his time to science. 
He spent his life in Gottingen in the midst of continuous 
work. In 1828 he went to Berlin to attend a meeting of 
scientists, but after this he never again left Gottingen, except 
in 1854, when a railroad was opened between Gottingen and 
Hanover. He had a strong will, and his character showed 
a curious mixture of self-conscious dignity and child-like 
simplicity. He was little communicative, and at times 
morose. 

A new epoch in the theory of numbers dates from the publi- 
cation of his Disquisitiones Arithmeticcey Leipzig, 1801. The 
beginning of this work dates back as far as 1795. Some of its 
results had been previously given by Lagrange and Euler, but 
were reached independently by Gauss, who had gone deeply 
into the subject before he became acquainted with the writ- 
ings of his great predecessors. The Disquisitiones Arithmeticce 



THBORY OF NUMBERS. 865 

was already in print when Legendre's TMorie des N(ymbres 
appeared. The great law of quadratic reciprocity, given in 
the fourth section of Gauss' work, a law which involves the 
whole theory of quadratic residues, was discovered by him by 
induction before he was eighteen, and was proved by him one 
year later. Afterwards he learned that Euler had imperfectly 
enunciated that theorem, and that Legendre had attempted 
to prove it, but met with apparently insuperable difficulties. 
In the fifth section Gauss gave a second proof of this '< gem " 
of higher arithmetic. In 1808 followed a third and fourth 
demonstration; in 1817, a fifth and sixth. Ko wonder that 
he felt a personal attachment to this theorem. Proofs were 
given also by Jacobi, Eisensteiu, Liouville, Lebesgue, A. 
Genocchi, Rummer, M. A. Stem, Chr. Zeller, Kronecker, 
Bouniakowsky, E. Schering, J. Petersen, Voigt, E. Busche, 
and Th. Pepin.^ The solution of the problem of the repre- 
sentation of numbers by binary quadratic forms is one of the 
great achievements of Gauss. He created a new algorithm by 
introducing the theory of congruences. The fourth section 
of the Disquisitiones Arithmeticce, treating of congruences of 
the second degree, and the fifth section, treating of quadratic 
forms, were, until the time of Jacobi, passed over with universal 
neglect, but they have since been the starting-point of a long 
series of important researches. The seventh or last section, 
developing the theory of the division of the circle, was received 
from the start with deserved enthusiasm, and has since been 
repeatedly elaborated for students. A standard work on 
Kreistheilung was published in 1872 by Paul Bachmann, then 
of Breslau. Gauss had planned an eighth section, which was 
omitted to lessen the expense of publication. His papers on 
the theory of numbers were not all included in his great treatise. 
Some of them were published for the first time after his death 
in his collected works (1863-1871). He wrote two memoirs on 



866 A HISTORY OF MATHBMATICS. • 

the theory of biquadratic residues (1825 and 1831), the second 
of which contains a theorem of biquadratic reciprocity. 

Gauss was led to astronomy by the discovery of the planet 
Ceres at Palermo in 1801. His determination of the elements 
of its orbit with sufficient accuracy to enable Olbers to redis- 
cover it, made the name of Gauss generally known. In 1809 
he published the Theoria motus corporum coelestium, which 
contains a discussion of the problems arising in the deter- 
mination of the movements of planets and comets from 
observations made on them imder any circumstances. In it 
are found four formulae in spherical trigonometry, now usually 
called "Gauss' Analogies," but which were published some- 
what earlier by Karl Brandon Mollweide of Leipzig (1774- 
1825), and earlier still by Jean Baptiste Joseph Delambre 
(1749-1822).** Many years of hard work were spent in the 
astronomical and magnetic observatory. He founded the 
Grerman Magnetic Union, with the object of securing con- 
tinuous observations at fixed times. He took part in geodetic 
observations, and in 1843 and 1846 wrote two memoirs, Ueber 
Gegenstiinde der hoheren Geodesie. He wrote on the attrac- 
tion of homogeneous ellipsoids, 1813. In a memoir on capil- 
lary attraction, 1833, he solves a problem in the calculus of 
variations involving the variation of a certain double integral, 
the limits of integration being also variable ; it is the earliest 
example of the solution of such a problem. He discussed the 
problem of rays of light passing through a system of lenses. 

Among Gauss' pupils were Christian Heinrich Schumacher, 
Christian Gerling, Friedrich Nicolai, August Ferdinand 
Mobius, Georg Wilhelm Struve, Johann Frantz Encke. 

Gauss' researches on the theory of numbers were the start- 
ing-point for a school of writers, among the earliest of whom 
was Jacobi. The latter contributed to Crelle's Journal an article 
on cubic residues, giving theorems without proofs. After the 



THEORY OF NUMBERS. 867 

publication of Oauss' paper on biquadratic residues, giving 
the law of biquadratic reciprocity, and his treatment of com- 
plex numbers, Jacobi found a similar law for cubic residues. 
By the theory of elliptical functions, he was led to beautiful 
theorems on the representation of numbers by 2, 4, 6, and 8 
squares. Next come the researches of Dirichlet, the expounder 
of Gauss, and a contributor of rich results of his own. 

Peter Gustay lejeune Dirichlet^ (1805-1859) was bom in 
IXiren, attended the gymnasium in Bonn, and then the 
Jesuit gymnasium in Cologne. In 1822 he was attracted to 
Paris by the names of Laplace, Legendre, Fourier, Poisson, 
Cauchy. The facilities for a mathematical education there 
were far better than in Germany, where Grauss was the only 
great figure. He read in Paris Gauss' DisquisiHones Arith- 
meticoe, a work which he never ceased to admire and study. 
Much in it was simplified by Dirichlet, and thereby placed 
within easier reach of mathematicians. His first memoir on 
the impossibility of certain indeterminate equations of the 
fifth degree was presented to the French Academy in 1825. 
He showed that Fermat's equation, af + y* = 2*, cannot exist 
when n = 5. Some parts of the analysis are, however, 
Legendre's. Euler and Lagrange had proved this when n is 3 
and 4, and Lam^ proved it when n = 7. Dirichlet's acquaint- 
ance with Fourier led him to investigate Fourier's series. He 
became decent in Breslau in 1827. In 1828 he accepted a 
position in Berlin, and finally succeeded Gauss at G5ttingen 
in 1855. The general principles on which depends the aver- 
age number of classes of binary quadratic forms of positive 
and negative determinant (a subject first investigated by 
Gauss) were given by Dirichlet in a memoir, Ueber die Bestim- 
ntung der mittleren Werthe in der Zahlentheorie, 1849. More 
recently F. Mertens of Graz has determined the asymptotic 
values of several numerical functions. Dirichlet gave some 



868 A HISTORY OF MATHEMATICS. 

attention to prime numbers. Gauss and Legendre had given 
expressions denoting approximately the asymptotic value 'of 
the number of primes inferior to a given limit, but it remained 
for Kiemann in his memoir, Ueher die AnzaM der PrimzaJUen 
unter einer gegehenen Ordsaej 1859, to give an investigation of 
the asymptotic frequency of primes which is rigorous. Ap- 
proaching the problem from a different direction, PatnuHj 
Tchebycheff, formerly professor in the University of St. Peters- 
burg (born 1821), established, in a celebrated memoir, Sur Us 
Nombres Premiers^ 1850, the existence of limits within which 
the sum of the logarithms of the primes P, inferior to a given 
number a?, must be comprised.® This paper depends on very 
elementary considerations, and, in that respect, contrasts 
strongly with Kiemann's, which involves abstruse theorems 
of the integral calculus. Poincar^'s papers, Sylvester's con- 
traction of TchebychefPs limits, with reference to the distri- 
bution of primes, and researches of J. Hadamard (awarded the 
Orand prix of 1892), are among the latest researches in this 
line. The enumeration of prime numbers has been undertaken 
at different times by various mathematicians. In 1877 the 
British Association began the preparation of factor-tables, 
under the direction of J. W. L. Glaisher. The printing, by 
the Association, of tables for the sixth million marked the 
completion of tables, to the preparation of which Germany, 
France, and England contributed, and which enable us to 
resolve into prime factors every composite number less than 
9,000,000. 

Miscellaneous contributions to the theory of numbers were 
made by Catichy, He showed, for instance, how to find all 
the infinite solutions of a homogeneous indeterminate equation 
of the second degree in three variables when one solution is 
given. He established the theorem that if two congruences, 
which have the same modulus, admit of a common solution, 



THEORY OF NUMBERS. 869 

the modulus is a divisor of their resultant. Joseph liouyille 
(1809-1882), professor at the College de France, investigated 
mainly questions on the theory of quadratic forms of two, and 
of a greater number of variables. Profound researches were 
instituted by Ferdinand Gotthold Eisenstein (1823-1852), of 
Berlin. Ternary quadratic forms had been studied somewhat 
by Gauss, but the extension from two to three indeterminates 
was the work of Eisenstein who, in his memoir. Neue Theo- 
reme der hoheren Arithmetik, defined the ordinal and generic 
characters of ternary quadratic forms of uneven determinant; 
and, in case of definite forms, assigned the weight of any order 
or genus. But he did not publish demonstrations of his re- 
sults. In inspecting the theory of binary cubic forms, he was 
led to the discovery of the first covariant ever considered in 
analysis. He showed that the series of theorems, relating to 
the presentation of numbers by sums of squares, ceases when 
the number of squares surpasses eight. Many of the proofs 
omitted by Eisenstein were supplied by Henry Smith, who 
was one of the few Englishmen who devoted themselves to the 
study of higher arithmetic. 

Henry John Stephen Smith'*' (1826-1883) was bom in Lon- 
don, and educated at Kugby and at Balliol College, Oxford. 
Before 1847 he travelled much in Europe for his health, 
and at one time attended lectures of Arago in Paris, but 
after that year he was never absent from Oxford for a 
single term. In 1861 he was elected Savilian professor of 
geometry. His first paper on the theory of numbers appeared 
in 1855. The results of ten years' study of everything pub- 
lished on the theory of numbers are contained in his Reports 
which appeared in the British Association volumes from 1859 
to 1865. These reports are a model of clear and precise 
exposition and perfection of form. They contain much orig- 
inal matter, but the chief results of his own discoveries were 



870 A HISTORY OF MATHBMATICS. 

printed in the Philosophical Transactions for 1861 and 1867. 
They treat of linear indeterminate equations and congraences, 
and of the orders and genera of ternary quadratic forms. He 
established the principles on which the extension to the gen- 
eral case of n indeterminates of quadratic forms depends. 
He contributed also two memoirs to the Proceedings of the 
Royal Society of 1864 and 1868, in the second of which he 
remarks that the theorems of Jacobi, Eisenstein, and Liou- 
ville, relating to the representation of numbers by 4, 6, 8 
squares, and other simple quadratic forms are deducible by a 
uniform method from the principles indicated in his paper. 
Theorems relating to the case of 5 squares were given by 
Eisenstein, but Smith completed the enunciation of them, and 
added the corresponding theorems for 7 squares. The solu- 
tion of the cases of 2, 4, 6 squares may be obtained by elliptic 
functions, but when the number of squares is odd, it involves 
processes peculiar to the theory of numbers. This class of 
theorems is limited to 8 squares, and Smith completed the 
group. In ignorance of Smith's investigations, the French 
Academy offered a prize for the demonstration and comple- 
tion of Eisenstein's theorems for 5 squares. This Smith had 
accomplished fifteen years earlier. He sent in a dissertation 
in 1882, and next year, a month after his death, the prize was 
awarded to him, another prize being also awarded to H. Min- 
kowsky of Bonn. The theory of numbers led Smith to the 
study of elliptic functions. He wrote also on modern geome- 
try. His successor at Oxford was J. J. Sylvester. 

Ernst Eduard Kummer (1810-1893), professor in the Uni- 
versity of Berlin, is closely identified with the theory of num- 
bers. Dirichlet's work on complex numbers of the form a4-i6, 
introduced by Gauss, was extended by him, by Eisenstein, 
and Dedekind. Instead of the equation x* — 1 = 0, the roots 
of which yield Gauss' units, Eisenstein used the equation 



THEORY OF NUMBERS. 871 

a^ — 1 = and complex numbers a + 6p (p being a cube root 
of unity )^ the theory of which resembles that of Gktuss' num- 
bers. Kummer passed to the general case x" — 1 = and got 
complex numbers of the form a^aiAi^-afAf^a^-^"'; 
where a< are whole real numbers^ and At roots of the above 
equation.'* Euclid's theory of the greatest common divisor 
is not applicable to such complex numbers, and their prime 
factors cannot be defined in the same way as prime factors of 
common integers are defined. In the effort to overcome this 
difficulty, Kummer was led to introduce the conception of 
'4deal numbers." These ideal numbers have been applied by 
O. Zolotareff of St Petersburg to the solution of a problem 
of the integral calculus, left unfinished by Abel (LiouviU^a 
Journal, Second Series, 1864, Vol. IX.). Julias Wilhelm Ridurd 
Dedekind of Braunschweig (born 1831) has given in the second 
edition of Dirichlet's Vorlesungen ilber Zahlentheorie a new 
theory of complex numbers, in which he to some extent 
deviates from the course of Kummer, and avoids the use of 
ideal numbers. Dedekind has taken the roots of any irreduci- 
ble equation with integral coefficients as the units for his com- 
plex numbers. Attracted by Kummer's investigations, his 
pupil, Leopold Kronecker (1823-1891) made researches which 
he applied to algebraic equations. 

On the other hand, efforts have been made to utilise in the 
theory of numbers the results of the modern higher algebra. 
Following up researches of Hermite, Paul Buchmmin of Mfinster 
investigated the arithmetical formula which gives the auto- 
morphics of a ternary quadratic form.* The problem of the 
equivalence of two positive or definite ternary quadratic forms 
was solved by L. Seeber ; and that of the arithmetical auto- 
morphics of such forms, by Eisenstein. The more difficult prob- 
lem of the equivalence for indefinite ternary forms has been 
investigated by Edward Selling of Wiirzburg. On quadratic 



872 A UISTOBY OF MATHBMATICS. 

forms of four or more indeterminates little has yet been done. 
Hermite showed that the number of non-equivalent classes of 
quadratic forms having integral coefficients and a given dis- 
criminant is Bmie, while Zolotareff and A. N. Korkine, both 
of St. Petersburg, investigated the minima of positive quadratic 
forms. In connection with binary quadratic forms. Smith 
established the theorem that if the joint invariant of two 
properly primitive forms vanishes, the determinant of either 
of them is represented primitively by the duplicate of the 
other. 

The interchange of theorems between arithmetic and algebra 
is displayed in the recent researches of J. W. L. Glaisher 
of Trinity College (born 1848) and Sylvester. Sylvester gave 
a Constructive Theory of Partitions, which received additions 
from his pupils, F. Franklin and G. S. Ely. 

The conception of "number" has been much extended in 
our time. With the Greeks it included only the ordinary 
positive whole numbers ; Diophantus added rational fractions 
to the domain of numbers. Later negative numbers and 
imaginaries came gradually to be recognised. Descartes fully 
grasped the notion of the negative ; Gauss, that of the imagi- 
nary. With Euclid, a ratio, whether rational or irrational, was 
not a number. The recognition of ratios and irrationals as 
numbers took place in the sixteenth century, and found expres- 
sion with Newton. By the ratio method, the continuity of the 
real number system has been based on the continuity of space, 
but in recent time three theories of irrationals have been 
advanced by Weierstrass, J. W. R. Dedekind, G. Cantor, and 
Heine, which prove the continuity of numbers without borrow- 
ing it from space. They are based on the definition of numbers 
by regular sequences, the use of series and limits, and some 
new mathematical conceptions. 



APPL1£D MATHEMATICS. 878 



APPLIED MATHEMATICS. 

Notwithstanding the beautiful developments of celestial 
mechanics reached by Laplace at the close of the eighteenth 
century, there was made a discovery on the first day of the 
present century which presented a problem seemingly beyond 
the power of that analysis. We refer to the discovery of Ceres 
by Piazzi in Italy, which became known in Germany just after 
the philosopher Hegel had published a dissertation proving a 
priori that such a discovery could not be made. From the 
positions of the planet observed by Piazzi its orbit could not 
be satisfactorily calculated by the old methods, and it remained 
for the genius of Gauss to devise a method of calculating 
elliptic orbits which was free from the assumption of a small 
eccentricity and inclination. Gauss' method was developed 
further in his Theoria Motus. The new planet was re-dis- 
covered with aid of Gauss' data by Olbers, an astronomer 
who promoted science not only by his own astronomical 
studies, but also by discerning and directing towards astro- 
nomical pursuits the genius of Bessel. 

Friedrich Wilhelm Bessel'^ (1784-1846) was a native of 
Minden in Westphalia. Fondness for figures, and a distaste 
for Latin grammar led him to the choice of a mercantile 
career. In his fifteenth year he became an apprenticed clerk 
in Bremen, and for nearly seven years he devoted his days to 
masti'ring the details of his business, and part of his nights to 
study. Hoping some day to become a supercargo on trading 
expeditions, he became interested in observations at sea. With 
a sextant constructed by him and an ordinary clock he deter- 
mined the latitude of Bremen. His success in this inspired 
him for astronomical study. One work after another was 
mastered by him, unaided, during the hours snatched from 



374 A HISTORY OF MATHEMATICS. 

sleep. From old observations he calculated the orbit of 
Halley's comet. Bessel introduced himself to Olbers, and 
submitted to him the calculation, which Olbers immediately 
sent for publication. Encouraged by Olbers, Bessel turned 
his back to the prospect of affluence, chose poverty and the 
stars, and became assistant in J. H. Schroter's observatory at 
Lilienthal. Four years later he was chosen to superintend 
the construction of the new observatory at Konigsberg.** In 
the absence of an adequate mathematical teaching force, Bessel 
was obliged to lecture on mathematics to prepare students for 
astronomy. He was relieved of this work in 1825 by the 
arrival of Jacobi. We shall not recount the labours by which 
Bessel earned the title of founder of modern practical astron- 
omy and geodesy. As an observer he towered far above 
Gauss, but as a mathematician he reverently bowed before the 
genius of his great contemporary. Of Bessel's papers, the one 
of greatest mathematical interest is an ^^ Untersuchung des 
TJieils der planetarischen Storungen, welcher aus der Bewegung 
der Sonne ensteht^^ (1824), in which he introduces a class of 
transcendental functions, Jn{^)j much used in applied mathe- 
matics, and known as "BessePs functions." He gave their 
principal properties, and constructed tables for their eval- 
uation. Recently it has been observed that BessePs func- 
tions appear much earlier in mathematical literature.* 
Such functions of the zero order occur in papers of Daniel 
Bernoulli (1732) and Euler on vibration of heavy strings sus- 
pended from one end. All of BessePs functions of the first 
kind and of integral orders occur in paper by Euler (1764) on 
the vibration of a stretched elastic membrane. In 1878 Lord 
Rayleigh proved that BesseVs functions are merely particular 
cases of Laplace's functions. J. W. L. Glaisher illustrates 
bv BessePs functions his assertion that mathematical branches 
growing out of physical inquiries as a rule " lack the easy flow 



APPLIED MATHEMATICS. 876 

or homogeneity of form which is characteristic of a mathemati- 
cal theory properly so called." These functions have been 
studied by C. Th. Anger of Danzig, 0. Schldmilch of Dresden, 
R. Lipschitz of Bonn (bom 1832), Carl Neumann of Leipzig 
(bom 1832), Eugen Lommel of Leipzig, I. Todhunter of St. 
John's College, Cambridge. 

Prominent among the successors of Laplace are the follow- 
ing: SimSon Denis Poisaon (1781-1840), who wrote in 1808 
a classic M&moire sur lea in^gdlitis siculaires des moyena mouve- 
ments des plan^tes, Oiovanni Antonio Amadeo Plana (1781- 
1864) of Turin, a nephew of Lagrange, who published in 1811 
a Memoria sulla teoria deW attrazione degli sferoidi eUitici, and 
contributed to the theory of the moon. Peter Andreas Hansen 
(1795-1874) of Grotha, at one time a clockmaker in Tondem, 
then Schumacher's assistant at Altona, and finally director of 
the obseryatory at Grotha, wrote on various astronomical sub- 
jects, but mainly on the lunar theory, which he elaborated in 
his work Fundamenta nova investigationes orbital verxE quam 
Luna perlustrat (1838), and in subsequent investigations 
embracing extensive lunar tables. George Biddel Airy (1801- 
1892), royal astronomer at Greenwich, published in 1826 his 
Mathematical Tracts on the Lunar and Planetary Theories. 
These researches have since been greatly extended by him. 
August Ferdinand Mobius (1790-1868) of Leipzig wrote, in 1842, 
Elemente der Mechanik des Himmels, Urbain Jean Joseph Le 
Verrier (1811-1877) of Paris wrote, the Recherches Astrono- 
miquesy constituting in part a new elaboration of celestial 
mechanics, and is famous for his theoretical discovery of 
Neptune. John Couch Adams (1819-1892) of Cambridge 
divided with Lc Verrier the honour of the mathematical dis- 
covery of Neptune, and pointed out in 1853 that Laplace's 
explanation of the secular acceleration of the moon's mean 
motion accounted for only half the observed acceleration. 



876 A HISTORY OF MATHEMATICS. 

Charles Eugene Delaonay (born 1816, and drowned off Cher- 
bourg iu 1872), professor of mechanics at the Sorbonne in 
Paris, explained most of the remaining acceleration of the 
moon, unaccounted for by Laplace's theory as corrected by 
Adams, by tracing the effect of tidal friction, a theory 
previously suggested independently by Kant, Robert Mayer, 
and William Ferrel of Kentucky. George Howard Darwin of 
Cambridge (born 1845) made some very remarkable inves- 
tigations in 1879 on tidal friction, which trace with great 
certainty the history of the moon from its origin. He has 
since studied also the effects of tidal friction upon other 
bodies in the solar system. Criticisms on some parts of his 
researches have been made by James Nolan of Victoria. Simon 
Newcomb (born 1835), superintendent of the Nautical Almanac 
at Washington, and professor of mathematics at the Johns 
Hopkins University, investigated the errors in Hansen's tables 
of the moon. For the last twelve years the main work of the 
C7. S. Nautical Almanac office has been to collect and discuss 
data for new tables of the planets which will supplant the 
tables of Le Verrier. (?. W, Hill of that office has contributed 
an elegant paper on certain possible abbreviations in the com- 
putation of the long-period of the moon's motion due to the 
direct action of the planets, and has made the most elaborate 
determination yet undertaken of the inequalities of the moon's 
motion due to the figure of the earth. He has also computed 
certain lunar inequalities due to the action of Jupiter. 

The mathematical discussion of Saturn's rings was taken up 
first by Laplace, who demonstrated that a homogeneous solid 
ring could not be in equilibrium, and in 1851 by B. Peirce, 
who proved their non-solidity by showing that even an irregu- 
lar solid ring could not be in equilibrium about Saturn. The 
mechanism of these rings was investigated by James Clerk 
Maxwell in an essay to which the Adams prize was awarded. 



APPLIED MATHEMATICS. 877 

He concluded that they consisted of an aggregate of uncon- 
nected particles. 

The problem of three bodies has been treated in various 
ways since the time of Lagrange, but no decided advance 
towards a more complete algebraic solution has been made, 
and the problem stands substantially where it was left by him. 
He had made a reduction in the differential equations to the 
seventh order. This was elegantly accomplished in a different 
way by Jacobi in 1843. R, Radau (Comptes Rendus, LXVII., 
1868, p. 841) and AU^gret {JoumcU de Math4matiqueSy 1875, 
p. 277) showed that the reduction can be performed on the 
equations in their original form. Noteworthy transformations 
and discussions of the problem have been given by J. L. F. 
Bertrand, by Emile Bour (1831-1866) of the Polytechnic School 
in Paris, by Mathieu, Hesse, J. A. Serret. H. Bruns of Leipzig 
has shown that no advance in the problem of three or of n 
bodies may be expected by algebraic integrals, and that we 
must look to the modern theory of functions for a complete 
solution (Acta Math,, XL, p. 43).* 

Among valuable text-books on mathematical astronomy rank 
the following works : Manual of Spherical and Practical Astron- 
omy by Chauvenet (1863), Practical and Spherical Astronomy 
by Robert Main of Cambridge, Theoretical Astronomy by James 
C. Watson of Ann Arbor (1868), Trait4 4Umentaire de M^cor 
nique CMeste of H. Resal of the Polytechnic School in Paris, 
Cours d' Astronomic de VEcole Polytechnique by Faye, Trait4 
de M^canique Celeste by Tisserand, Lehrbuch der Bahnbestimr 
mung by T. Oppolzery Mathematische Theorien der Planetenr 
bewegung by O. Dziobeky translated into English by M. W. 
Harrington and W. J. Hussey. 

During the present century we have come to recognise the 
advantages frequently arising from a geometrical treatment of 
mechanical problems. To Poinsot, Ghasles, and Mdbios we 



378 A HISTORY OF MATHEMATICS. 

• 

owe the most important developments made in geometrieal 
mechanics. Lonis Poinsot (1777-1859), a graduate of the 
Polytechnic School in Paris, and for many years member of 
the superior council of public instruction, published in 1804 
his Elements de StcUiqtte, This work is remarkable not only 
as being the earliest introduction to synthetic mechanics, but 
also as containing for the first time the idea of couples, which 
was applied by Poinsot in a publication of 1834 to the theory 
of rotation. A clear conception of the nature of rotary 
motion was conveyed by Poinsot's elegant geometrical repre- 
sentation by means of an ellipsoid rolling on a certain fixed 
plane. This construction was extended by Sylvester so as 
to measure the rate of rotation of the ellipsoid on the plane. 

A particular class of dynamical problems has recently been 
treated geometrically by Sir Robert Stawell Ball, formerly 
astronomer royal of Ireland, now Lowndean Professor of 
Astronomy and Geometry at Cambridge. His method is given 
in a work entitled Theory of Screws, Dublin, 1876, and in 
subsequent articles. Modern geometry is here drawn upon, 
as was done also by Clifford in the related subject of Bi- 
quaternions. Arthur Buchheim of Manchester (1859-1888), 
showed that Grassmann's Ausdehnungslehre supplies all the 
necessary materials for a simple calculus of screws in elliptic 
space. Horace Lamb applied the theory of screws to the ques- 
tion of the steady motion of any solid in a fluid. 

Advances in theoretical mechanics, bearing on the in- 
tegration and the alteration in form of dynamical equations, 
were made since Lagrange by Poisson, William Rowan Hamil- 
ton, Jacobi, Madame Kowalevski, and others. Lagrange had 
established the "Lagrangian form" of the equations of 
motion. He had given a theory of the variation of the 
arbitrary constants which, however, turned out to be less 
fruitful in results than a theory advanced by Poisson.* Pols- 



APPLIBD MATHEMATICS. 879 

son's theory of the variation of the arbitrary constants and 
the method of integration thereby afforded marked the first 
onward step since Lagrange. Then came the researches of 
Sir William Kowan Hamilton. His discovery that the inte- 
gration of the dynamic differential equations is connected with 
the integration of a certain partial differential equation of the 
first order and second degree, grew out of an attempt to deduce, 
by the undulatory theory, results in geometrical optics previ- 
ously based on the conceptions of the emission theory. The 
Philosophical Transactions of 1833 and 1834 contain Hamil- 
ton's papers, in which appear the first applications to me- 
chanics of the principle of varying action and the characteristic 
function, established by him some years previously. The 
object which Hamilton proposed to himself is indicated by 
the title of his first paper, viz. the discovery of a function 
by means of which all integral equations can be actually 
represented. The new form obtained by him for the equation 
of motion is a result of no less importance than that which 
was the professed object of the memoir. Hamilton's method 
of integration was freed by Jacobi of an unnecessary complica- 
tion, and was then applied by him to the determination of a 
geodetic line on the general ellipsoid. With aid of elliptic co- 
ordinates Jacobi integrated the partial differential equation 
and expressed the equation of the geodetic in form of a 
relation between two Abelian integrals. Jacobi applied to 
differential equations of dynamics the theory of the ultimate 
multiplier. The differential equations of dynamics are only 
one of the classes of differential equations considered by 
Jacobi. Dynamic investigations along the lines of Lagrange, 
Hamilton, and Jacobi were made by Liouville, A. Desboves, 
Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin, 
Brioschi, leading up to the development of the theory of a 
system of canonical integrals. 



880 A HISTORY OF MATHEMATICS. 

An important addition to the theory of the motion of a solid 
body about a fixed point was made by Madame Sophie de 
Kowaleyski^ (1853-1891), who discovered a new case in which 
the differential equations of motion can be integrated. By 
the use of theta-functions of two independent variables she 
furnished a remarkable example of how the modem theory of 
functions may become useful in mechanical problems. She 
was a native of Moscow, studied under Weierstrass, obtained 
the doctor's degree at Gottingen, and from 1884 until her 
death was professor of higher mathematics at the University 
of Stockholm. The research above mentioned received the 
Bordin prize of the French Academy in 1888, which was 
doubled on account of the exceptional merit of the paper. 

There are in vogue three forms for the expression of the 
kinetic energy of a dynamical system: the Lagrangian, the 
Hamiltonian, and a modified form of Lagrange's equations in 
which certain velocities are omitted. The kinetic energy 
is expressed in the first form as a homogeneous quadratic 
function of the velocities, which are the time-variations of the 
co-ordinates of the system; in the second form, as a homo- 
geneous quadratic function of the momenta of the system ; 
the third form, elaborated recently by Edward John Routh 
of Cambridge, in connection with his theory of " ignoration of 
co-ordinates," and by A. B. Basset, is of importance in hydro- 
dynamical problems relating to the motion of perforated solids 
in a liquid, and in other branches of physics. 

In recent time great practical importance has come to be 
attached to the principle of mechanical similitude. By it one 
can determine from the performance of a model the action of 
the machine constructed on a larger scale. The principle was 
first enunciated by Newton (Principiay Bk. II., Sec. VIII., 
Prop. 32), and was derived by Bei-trand from the principle 
of virtual velocities. A corollary to it, applied in ship-build- 



APPLIED MATHEMATICS. 881 

ingy goes by the name of William Froude's law, but was enun- 
ciated also by Reech. 

The present problems of dynamics differ materially from 
those of the last century. The explanation of the orbital and 
axial motions of the heavenly bodies by the law of universal 
gravitation was the great problem solved by Clairaut, Euler, 
lyAlembert, Lagrange, and Laplace. It did not involve the 
consideration of frictional resistances. In the present time 
the aid of dynamics has beeu invoked by the physical 
sciences. The problems there arising are often complicated 
by the presence of friction. Unlike astronomical problems of 
a century ago, they refer to phenomena of matter and motion 
that are usually concealed from direct observation. The great 
pioneer in such problems is Lord Kelvin. While yet an 
undergraduate at Cambridge, during holidays spent at the 
seaside, he entered upon researches of this kind by working 
out the theory of spinning tops, which previously had been 
only partially explained by Jellet in his Treatise on the Theory 
of Friction (1872), and by Archibald Smith. 

Among standard works on mechanics are Jacobi's Vorlesun- 
gen iiber Dynamik, edited by Clebsch, 1866 ; Kirchhoff's Vorle- 
sungen iiber mathematische Physiky 1876 ; Benjamin Peirce's 
Analytic Mechanics^ 1855; Somoff's TJieoretische Mechanik, 
1879; Tait and Steele's Dynamics of a Particle, 1856; Bfinchin's 
Treatise on Statics; Routh's Dynamics of a System of Rigid 
Bodies; Sturm's Cours de Micainque de VEcole Polytechnique. 

The equations which constitute the foundation of the theory 
of fluid motion were fully laid down at the time of Lagrange, 
but the solutions actually worked out were few and mainly 
of the irrotational type. A powerful method of attacking 
problems in fluid motion is that of images, introduced in 1843 
by George Gabriel Stokes of Pembroke College, Cambridge. 
It received little attention until Sir William Thomson's dis- 



382 A HISTORY OF MATHEMATICS. 

coveiy of electrical images, whereupon the theory was extended 
by Stokes, Hicks, and Lewis. In 1849, Thomson gave the 
maximum and minimum theorem peculiar to hydrodynamics, 
which was afterwards extended to dynamical problems in 
general. 

A new epoch in the progress of hydrodynamics was created, 
in 1856, by Helmholtz, who worked out remarkable properties 
of rotational motion in a homogeneous, incompressible fluid, 
devoid of viscosity. He showed that the vortex filaments in 
such a medium may possess any number of knottings and twist- 
ings, but are either endless or the ends are in the free surface 
of the medium ; they are indivisible. These results suggested 
to Sir William Thomson the possibility of founding on them a 
new form of the atomic theory, according to which every atom 
is a vortex ring in a non-frictional ether, and as such must be 
absolutely permanent in substance and duration. The vortex- 
atom theory is discussed by J. J. Thomson of Cambridge 
(born 1856) in his classical treatise on the Motion of Vortex 
Rings, to which the Adams Prize was awarded in 1882. 
Papers on vortex motion have been published also by Horace 
Lamb, Thomas Craig, Henry A. Rowland, and Charles Chree. 

The subject of jets was investigated by Helmholtz, Kirch- 
hoff. Plateau, and Rayleigh ; the motion of fluids in a fluid by 
Stokes, Sir W. Thomson, Kopcke, Greenhill, and Lamb ; the 
theory of viscous fluids by Navier, Poisson, Saint-Venant, 
Stokes, 0. E. Meyer, Stefano, Maxwell, Lipschitz, Craig, 
Helmholtz, and A. B. Basset. Viscous fluids present great 
difficulties, because the equations of motion have not the same 
degree of certainty as in perfect fluids, on account of a defi- 
cient theory of friction, and of the difficulty of connecting 
oblique pressures on a small area with the differentials of the 
velocities. 

Waves in liquids have been a favourite subject with Eng- 



APPLIED MATHEMATI08. 888 

lish mathematicians. The early inquiries of Poisson and 
Gauchy were directed to the iuvestigatiou of waves produced 
by disturbing causes acting arbitrarily on a small portion 
of the fluid. The velocity of the long wave was given 
approximately by Lagrange in 1786 in case of a channel of 
rectangular cross-section, by Green in 1839 for a channel of 
triangular section, and by P. Kelland for a channel of any 
uniform section. Sir Greorge B. Airy, in his treatise on Tides 
and Waves, discarded mere approximations, and gave the exact 
equation on which the theory of the long wave in a channel of 
uniform rectangular section depends. But he gave no general 
solutions. J. McGowan of University College at Dundee 
discusses this topic more fully, and arrives at exact and 
complete solutions for certain cases. The most important 
application of the theory of the long wave is to the explana- 
tion of tidal phenomena in rivers and estuaries. 

The mathematical treatment of solitary waves was first 
taken up by S. Earnshaw in 1845, then by Stokes ; but the first 
sound approximate theory was given by J. Boussinesq in 1871, 
who obtained an equation for their form, and a value for the 
velocity in agreement with experiment. Other methods of 
approximation were given by Rayleigh and J. McCowan. In 
connection with deep-water waves, Osborne Reynolds gave in 
1877 the dynamical explanation for the fact that a group 
of such waves advances with only half the rapidity of the 
individual waves. 

The solution of the problem of the general motion of an 
ellipsoid in a fluid is due to the successive labours of Green 
(1833), Clebsch (1856), and Bjerknes (1873). The free 
motion of a solid in a liquid has been investigated by W. 
Thomson, Kirchhoff, and Horace Lamb. By these labours, the 
motion of a single solid in a fluid has come to be pretty well 
understood, but the case of two solids in a fluid is not devel- 



884 A HISTOBY OF MATHEMATICS. 

oped SO folly. The problem has been attacked by W. M. 
Hicks. 

The determination of the period of oscillation of a rotating 
liquid spheroid has important bearings on the question of the 
origin of the moon. G. H. Darwin's investigations thereon, 
viewed in the light of Kiemann's and Poincar^'s researches, 
seem to disprove Laplace's hypothesis that the moon separated 
from the earth as a ring, because the angular velocity was too 
great for stability ; Darwin finds no instability. 

The explanation of the contracted vein has been a point of 
much controversy, but has been put in a much better light by 
the application of the principle of momentum, originated by 
Eroude and Kayleigh. Eayleigh considered also the reflection 
of waves, not at the surface of separation of two uniform 
media, where the transition is abrupt, but at the confines of 
two media between which the transition is gradual. 

The first serious study of the circulation of winds on the 
earth's surface was instituted at the beginning of the second 
quarter of this century by U, W. Dov4, William C. Redjield^ and 
James P. Espy, followed by researches of W. Reid, Piddington, 
and Elia^ Loomis. But the deepest insight into the wonder- 
ful correlations that exist among the varied motions of the 
atmosphere was obtained by William Ferrel (1817-1891). He 
was born in Fulton County, Pa., and brought up on a farm. 
Though in unfavourable surroundings, a burning thirst for 
knowledge spurred the boy to the mastery of one branch after 
another. He attended Marshall College, Pa., and graduated 
in 1844 from Bethany College. While teaching school he 
became interested in meteorology and in the subject of tides. 
In 1856 he wrote an article on " the winds and currents of the 
ocean." The following year he became connected with the 
Nautical Almanac, A mathematical paper followed in 1868 
on "the motion of fluids and solids relative to the earth's 



APPLIED MATHEMATICS. 386 

sarfaoe." The subject was extended afterwards so as to 
embrace the mathematical theory of cjclonesy tornadoes, 
water-spouts, etc. In 1885 appeared his Recent Advances in 
Meteorology, In the opinion of a leading European meteor- 
ologist (Julius Hann of Vienna), Ferrel has ''contributed more 
to the advance of the physics of the atmosphere than any 
other living physicist or meteorologist." 

Ferrel teaches that the air flows in great spirals toward the 
poles, both in the upper strata of the atmosphere and on the 
earth's surface beyond the 30th degree of latitude; while 
the return current blows at nearly right angles to the above 
spirals, in the middle strata as well as on the earth's surface, 
in a zone comprised between the parallels 30^ N. and 30^ S. The 
idea of three superposed currents blowing spirals was first 
advanced by James Thomson, but was published in very 
meagre abstract. 

Ferrel's views have given a strong impulse to theoretical 
research in America, Austria, and Genpany. Several objec- 
tions raised against his argument have been abandoned, or 
have been answered by W. M. Davis of Harvard. The mathe- 
matical analysis of F. Waldo of Washington, and of others, 
has further confirmed the accuracy of the theory. The trans- 
port of Krakatoa dust and observations made on clouds point 
toward the existence of an upper east current on the equator, 
and Fernter has mathematically deduced from Ferrel's theory 
the existence of such a current. 

Another theory of the general circulation of the atmosphere 
was propounded by Werner Siemens of Berlin, in which an 
attempt is made to apply thermodynamics to aerial currents. 
Important new points of view have been introduced recently 
by Helniholtz, who concludes that when two air currents blow 
one above the other in different directions, a system of air 
waves must arise in the same way as waves are formed on the 



386 A HISTORY OF MATHEMATICS. 

sea. He and A. Oberbeck showed that when the waves on the 
sea attain lengths of from 16 to 33 feet, the air waves must 
attain lengths of from 10 to 20 miles, and proportional depths. 
Superposed strata would thus mix more thoroughly, and their 
energy would be partly dissipated. From hydrodynaniical 
equations of rotation Helmholtz established the reason why 
the observed velocity from equatorial regions is much less in 
a latitude of, say, 20® or 30®, than it would be were the move- 
ments unchecked. 

About 1860 acoustics began to be studied with renewed 
zeal. The mathematical theory of pipes and vibrating strings 
had been elaborated in the eighteenth century by Daniel Ber- 
noulli, D'Alembert, Euler, and Lagrange. In the first part of 
the present century Laplace corrected Newton's theory on the 
velocity of sound in gases, Poisson gave a mathematical dis- 
cussion of torsional vibrations ; Poisson, Sophie Grermain, and 
Wheatstoue studied Chladni's figures ; Thomas Young and the 
brothers Weber developed the wave-theory of sound. Sir J. 
F. W. Herschel wrote on the mathematical theory of sound for 
the Encyclopcedia Metropolitana, 1845. Epoch-making were 
Helmholtz's experimental and mathematical researches. In 
his hands and Eayleigh's, Fourier's series received due 
attention. Helmholtz gave the mathematical theory of beats, 
difference tones, and summation tones. Lord Rayleigh (John 
William Strutt) of Cambridge (born 1842) made extensive 
mathematical researches in acoustics as a part of the theory of 
vibration in general. Particular mention may be made of his 
discussion of the disturbance produced by a spherical obstacle 
on the waves of sound, and of phenomena, such as sensitive 
flames, connected with the instability of jets of fluid. In 1877 
and 1878 he published in two volumes a treatise on Tlie Theory 
of Sound. Other mathematical researches on this subject have 
been made in England by Donkin and Stokes. 



APPLIED MATHEMATICS. 887 

The theory of elasticity ^ belongs to this century. Before 
1800 no attempt had been made to form general equations for 
the motion or equilibrium of an elastic solid. Particular prob- 
lems had been solved by special hypotheses. Thus, James 
Bernoulli considered elastic laminae; Daniel Bernoulli and 
Euler investigated vibrating rods; Lagrange and Euler, the 
equilibrium of springs and columns. The earliest investiga- 
tions of this century, by Thomas Young (" Young's modulus of 
elasticity ") in England, J. Binet in France, and O. A. A. Plana 
in Italy, were chiefly occupied in extending and correcting the 
earlier labours. Between 1830 and 1840 the broad outline of the 
modern theory of elasticity was established. This was accom- 
plished almost exclusively by French writers, — Louis-Marie- 
Henri Navier (1785-1836), Poisson, Cauchy, Mademoiselle 
Sophie Germain (1776-1831), F^lix Savart (1791-1841). 

Sim^n Denis Poisson^ (1781-1840) was bom at Pithiviers. 
The boy was put out to a nurse, and he used to tell that when 
his father (a common soldier) came to see him one day, the 
nurse had gone out and left him suspended by a thin cord to a 
nail in the wall in order to protect him from perishing under 
the teeth of the carnivorous and unclean animals that roamed 
on the floor. Poisson used to add that his gymnastic efforts 
when thus suspended caused him to swing back and forth, and 
thus to gain an early familiarity with the pendulum, the study 
of which occupied him much in his maturer life. His father 
destined him for the medical profession, but so repugnant was 
this to him that he was permitted to enter the Polytechnic 
School at the age of seventeen. His talents excited the inter- 
est of Lagrange and Laplace. At eighteen he wrote a memoir 
on finite differences which was printed on the recommendation 
of Legendre. He soon became a lecturer at the school, and 
continued through life to hold various government scientific 
posts and professorships. He prepared some 400 publications, 



888 A HISTORY OF MATHEMATICS. 

mainly on applied mathematics. His TraU4 de M^canique^ 
2 vols., 1811 and 1833, was long a standard work. He wrote 
on the mathematical theory of heat, capillary action, proba- 
bility of judgment, the mathematical theory of electricity and 
magnetism, physical astronomy, the attraction of ellipsoids, 
definite integrals, series, and the theory of elasticity. He was 
considered one of the leading analysts of his time. 

His work on elasticity is hardly excelled by that of C^uc^Jy 
and second only to that of Saint- Venant. There is hardly a 
problem in elasticity to which he has not contributed, while 
many of his inquiries were new. The equilibrium and motion 
of a circular plate was first successfully treated by him. 
Instead of the definite integrals of earlier writers, he used 
preferably finite summations. Poisson's contour conditions 
for elastic plates were objected to by Gustav Elirchhoff of 
Berlin, who established new conditions. But Thomson and 
Tait in their Treatise on Natural Philosophy have explained 
the discrepancy between Poisson's and KirchhofPs boundary 
conditions, and established a reconciliation between them. 

Important contributions to the theory of elasticity were 
made by Cauchy. To him we owe the origin of the theory 
of stress, and the transition from the consideration of the 
force upon a molecule exerted by its neighbours to the con- 
sideration of the stress upon a small plane at a point. He 
anticipated Green and Stokes in giving the equations of iso- 
tropic elasticity with two constants. The theory of elasticity 
was presented by Gabrio Piola of Italy according to the prin- 
ciples of Lagrange's M^canique Analytique, but the superiority 
of this method over that of Poisson and Cauchy is far from 
evident. The influence of temperature on stress was first 
investigated experimentally by AVilhelm Weber of Gottingen, 
and afterwards mathematically by Duhamel, who, assuming 
Poisson's theory of elasticity, examined the alterations of 



APPLIED MATHEMATICS. 889 

form which the formulae undergo when we allow for changes 
of temperature. Weber was also the first to experiment 
on elastic after-strain. Other important experiments were 
made by different scientists, which disclosed a wider range 
of phenomena, and demanded a more comprehensive theory. 
Set was investigated by Grerstner (1756-1832) and Eaton 
Hodgkinson, while the latter physicist in England and Vicat 
(1786-1861) in France experimented extensively on absolute 
strength. Vicat boldly attacked the mathematical theories of 
flexure because they failed to consider shear and the time-ele- 
ment. As a result, a truer theory of flexure was soon pro- 
pounded by Saint- Venant. Poncelet advanced the theories of 
resilience and cohesion. 

Gabriel Lam^^ (1795-1870) was bom at Tours, and gradu- 
ated at the Polytechnic School. He was called to Russia 
with Clapeyron and others to superintend the construction of 
bridges and roads. On his return, in 1832, he was elected 
professor of physics at the Polytechnic School. Subsequently 
he held various engineering posts and professorships in Paris. 
As engineer he took an active part in the construction of the 
first railroads in France. Lam^ devoted his fine mathemati- 
cal talents mainly to mathematical physics. In four works : 
Le^ns 8ur les fonctions inverses des transcendarUes et les sur- 
faces isothermes; Surles coordonn^es curvilignes et leurs diverses 
applications; Sur la th^rie andlytique de la chaleur; Sur la 
thiorie mathhtioiique de VUasticiti des corps solides (1852) , and 
in various memoirs he displays fine analytical powers ; but a 
certain want of physical touch sometimes reduces the value of 
his contributions to elasticity and other physical subjects. In 
considering the temperature in the interior of an ellipsoid 
under certain conditions, he employed functions analogous to 
Laplace's functions, and known by the name of *' Lamp's func- 
tions." A problem in elasticity called by Lamp's name, viz. 



s? 



390 A HISTORY OF MATHElblATICS. 

to investigate the conditions for equilibrium of a spherical 
elastic envelope subject to a given distribution of load on 
the bounding spherical surfaces, and the determination of the 
resulting shifts is the only completely general problem on 
elasticity which can be said to be completely solved. He 
deserves much credit for his derivation and transformation 
of the general elastic equations, and for his application of 
them to double refraction. Bectangular and triangular mem- 
branes were shown by him to be connected with questions in 
the theory of numbers. The field of photo-elasticity was 
entered upon by Lam^, F. E. Neumann, Clerk Maxwell. 
Stokes, Wertheim, R. Clausius, Jellett, threw new light upon 
the subject of " rari-constancy " and " multi-constancy," which 
has long divided elasticians into two opposing factions. The 
uni-constant isotropy of Navier and Poisson had been ques- 
tioned by Cauchy, and was now severely criticised by Green 
and Stokes. 

Barre de Saint- Venant (1797-1886), ingenieur des ponts et 
chaussees, made it his life-work to render the theory of 
elasticity of practical value. The charge brought by practical 
engineers, like Vicat, against the theorists led Saint-Venant to 
place the theory in its true place as a guide to the practical 
man. Numerous errors committed by his predecessors were 
removed. He corrected the theory of flexure by the considera- 
tion of slide, the theory of elastic rods of double curvature by 
the introduction of the third moment, and the theory of tor- 
sion by the discovery of the distortion of the primitively 
plane section. His results on torsion abpund in beautiful 
graphic illustrations. In case of a rod, upon the side surfaces 
of which no forces act, he showed that the problems of flexure 
and torsion can be solved, if the end-forces are distributed 
over the end-surfaces by a definite law. Clebsch, in his 
Lehrbuch der Elasticitdt, 1862, showed that this problem is 



APPLIED MATHEBiATICS. 891 

reversible to the case of side-forces without end-forces. 
Clebsch® extended the research to very thin rods and to very 
thin plates. Saint- Venant considered problems arising in the 
scientific design of built-up artillery, and his solution of them 
differs considerably from Lamp's solution, which was popular- 
ised by Eankine, and much used by gun-designers. In Saint- 
Venant's translation into French of Clebsch's Elasticitdt, he 
develops extensively a double-suffix notation for strain and 
stresses. Though often advantageous, this notation is cum- 
brous, and has not been generally adopted. Karl Pearaon, 
professor in University College, London, has recently exam- 
ined mathematically the permissible limits of the application 
of the ordinary theory of flexure of a beam. 

The mathematical theory of elasticity is still in an unsettled 
condition. Not only are scientists still divided into two 
schools of " rari-constancy " and " multi-constancy," but differ- 
ence of opinion exists on other vital questions. Among the 
numerous modern writers on elasticity may be mentioned 
Emile Mathieu (1835-1891), professor at Besanqon, Maurice 
Levy of Paris, Charles Chree, superintendent of the Kew Ob- 
servatory, A. B. Basset, Sir William Thomson (Lord Kelvin) 
of Glasgow, J. Boussinesq of Paris, and others. Sir William 
Thomson applied the laws of elasticity of solids to the investi- 
gation of the earth's elasticity, which is an important element 
in the theory of ocean-tides. If the earth is a solid, then its 
elasticity co-operates with gravity in opposing deformation 
due to the attraction of the sun and moon. Laplace had 
shown how the earth would behave if it resisted deformation 
only by gravity. Lam^ had investigated how a solid sphere 
would change if its elasticity only came into play. Sir 
William Thomson combined the two results, and compared 
them with the actual deformation. Thomson, and afterwards 
G. H. Darwin, computed that the resistance of the earth to 



892 A HISTORY OF MATHEMATICS. 

tidal deformation is nearly as great as though it were of steel. 
This conclusion has been confirmed recently by Simon New- 
comb, from the study of the observed periodic changes in 
latitude. For an ideally rigid earth the period would be 360 
days, but if as rigid as steel, it would be 441, the observed 
period being 430 days. 

Among text-books on elasticity may be mentioned the works 
of Lam^, Clebsch, Winkler, Beer, Mathieu, W. J. Ibbetson, and 
F. Neumann, edited by 0. E. Meyer. 

Riemann's opinion that a science of physics only exists since 
the invention of differential equations finds corroboration even 
in this brief and fragmentary outline of the progress of mathe- 
matical physics. The undulatory theory of light, first ad- 
vanced by Huygens, owes much to the power of mathematics : 
by mathematical analysis its assumptions were worked out 
to their last consequences. Thomas Young"* (1773-1829) was 
the first to explain the principle of interfereuce, both of 
light and sound, and the first to bring forward the idea 
of transverse vibrations in light waves. Young's explana- 
tions, not being verified by him by extensive numerical calcu- 
lations, attracted little notice, and it was not until Augustin 
Fresnel (1788-1827) applied mathematical analysis to a much 
greater extent than Young had done, that the undulatory 
theory began to carry conviction. Some of Fresnel's mathe- 
matical assumptions were not satisfactory; hence Laplace, 
Poisson, and others belonging to the strictly mathematical 
school, at first disdained to consider the theory. By their 
opposition Fresnel was spurred to greater exertion. Arago 
was the first great convert made by Fresnel. When polarisa- 
tion and double refraction were explained by Yoimg and 
Fresnel, then Laplace was at last won over. Poisson drew 
from FresnePs formulae the seemingly paradoxical deduction 
that a small circular disc, illuminated by a luminous pointy 



APPLIED MATHEMATICS. 898 

must cast a shadow with a bright spot in the centre. But 
this was found to be in accordance with fact. The theory 
was taken up by another great mathematician, Hamilton, who 
from his formulae predicted conical refraction, verified experi- 
mentally by Lloyd. These predictions do not prove, however, 
that Fresnel's formulae are correct, for these prophecies might 
have been made by other forms of the wave-theory. The 
theory was placed on a sounder dynamical basis by the writ- 
ings of Cauchy, Biot, Green, C. Neumann, Kirchhoff, McCullagh, 
Stokes, Saint-Venant, Sarrau, Lorenz, and Sir William Thom- 
son. In the wave-theory, as taught by Green and others, the 
luminiferous ether was an incompressible elastic solid, for 
the reason that fluids could not propagate transverse vibra- 
tions. But, according to Green, such an elastic solid would 
transmit a longitudinal disturbance with infinite velocity. 
Stokes remarked, however, that the ether might act like 
a fluid in case of finite disturbances, and like an elastic solid 
in case of the infinitesimal disturbances in light propagation. 

Fresnel postulated the density of ether to be different in 
different media, but the elasticity the same, while C. Neumann 
and McCullagh assume the density uniform and the elasticity 
different in all substances. On the latter assumption the 
direction of vibration lies in the plane of polarisation, and not 
perpendicular to it, as in the theory of Fresnel. 

While the above writers endeavoured to explain all optical 
properties of a medium on the supposition that they arise 
entirely from difference in rigidity or density of the ether in 
the medium, there is another school advancing theories in 
which the mutual action between the molecules of the body 
and the ether is considered the main cause of refraction and 
dispersion.^^ The chief workers in this field are J. Boussinesq, 
W. Sellmeyer, Helmholtz, E. Lommel, E. Ketteler, W. Voigt, 
and Sir William Thomson in his lectures delivered at the 



394 A HISTORY OF MATHEMATICS. 

Johns Hopkins University in 1884. Neither this nor the 
first-named school succeeded in explaining all the phenomena. 
A third school was founded by Maxwell. He proposed the 
electro-magnetic theory, which has received extensive develop- 
ment recently. It will be mentioned again later. According 
to Maxwell's theory, the direction of vibration does not lie 
exclusively in the plane of polarisation, nor in a plane perpen- 
dicular to it, but something occurs in both planes — a magnetic 
vibration in one, and an electric in the other. Fitzgerald and 
Trouton in Dublin verified this conclusion of Maxwell by 
experiments on electro-magnetic waves. 

Of recent mathematical and experimental contributions to 
optics, mention must be made of H. A. Rowland's theory of 
concave gratings, and of A. A. Michelson's work on interfer- 
ence, and his application of interference methods to astro- 
nomical measurements. 

In electricity the mathematical theory and the measure- 
ments of Henry Cavendish (1731-1810), and in magnetism 
the measurements of Charles Augustin Coulomb (1736-1806), 
became the foundations for a system of measurement. For 
electro-magnetism the same thing was done by Andre Marie 
Ampere (1775-1836). The first complete method of measure- 
ment was the system of absolute measurements of terrestrial 
magnetism introduced by Gauss and Wilhelm Weber (1804- 
1891) and afterwards extended by Wilhelm Weber and F. 
Kohlrausch to electro-magnetism and electro-statics. In 1861 
the British Association and the Royal Society appointed a 
special commission with Sir William Thomson at the head, to 
consider the unit of electrical resistance. The commission 
recommended a unit in principle like W. Weber's, but greater 
than Weber's by a factor of 10^^°^ The discussions and labours 
on this subject continued for twenty years, until in 1881 a 
general agreement was reached at an electrical congress in Paris. 



APPLIED MATHEMATICS. 895 

A function of fundamental importance in the mathematical 
theories of electricity and magnetism is the " potential." It 
was first used by Lagrange in the determination of gravita- 
tional attractions in 1773. Soon after, Laplace gave the 
celebrated differential equation, 

da? dy* da? 

which was extended by Poisson by writing -^Airk in place of 
zero in the right-hand member of the equation, so that it 
applies not only to a point external to the attracting mass, 
but to any point whatever. The first to apply the potential 
function to other than gravitation problems was George Green 
(1793-1841). He introduced it into the mathematical theory 
of electricity and magnetism. Green was a self-educated man 
who started out as a baker, and at his death was fellow of 
Caius College, Cambridge. In 1828 he published by subscrip- 
tion at Nottingham a paper entitled Essay on the application 
of mathematical analysis to the theory of electricity and magne- 
tism. It escaped the notice even of English mathematicians 
until 1846, when Sir William Thomson had it reprinted in 
CreUe's Journal, vols. xliv. and xlv. It contained what is now 
known as " Green's theorem " for the treatment of potential. 
Meanwhile all of Green's general theorems had been re-dis- 
covered by Sir William Thomson, Chasles, Sturm, and Gauss. 
The term potential function is due to Green. Hamilton used 
the word force-function, while Gauss, who about 1840 secured 
the general adoption of the function, called it simply potential. 
Large contributions to electricity and magnetism have been 
made by William Thomson. He was bom in 1824 at Belfast, 
Ireland, but is of Scotch descent. He and his brother James 
studied in Glasgow. From there he entered Cambridge, and 
was graduated as Second Wrangler in 1845. William Thom- 



896 A HISTORY OF MATHEMATICS. 

son, Sylvester, Maxwell, Clifford, and J. J. Thomson are a gronp 
of great men who were Second Wranglers at Cambridge. At 
the age of twenty-two W. Thomson was elected professor of 
natural philosophy in the University of Glasgow, a position 
which he has held ever since. For his brilliant mathematical 
and physical achievements he was knighted, and in 1892 was 
made Lord Kelvin. His researches on the theory of potential 
are epoch-making. What is called "Dirichlet's principle" 
was discovered by him in 1848, somewhat earlier than by 
Dirichlet. We owe to Sir William Thomson new synthetical 
methods of great elegance, viz. the theory of electric images 
and the method of electric inversion founded thereon. By 
them he determined the distribution of electricity on a bowl, 
a problem previously considered insolvable. The distribution 
of static electricity on conductors had been studied before this 
mainly by Poisson and Plana. In 1845 F. E. Neumann of 
Konigsberg developed from the experimental laws of Lenz the 
mathematical theory of magneto-electric induction. In 1855 
W. Thomson predicted by mathematical analysis that the dis- 
charge of a Leyden jar through a linear conductor would in 
certain cases consist of a series of decaying oscillations. This 
was first established experimentally by Joseph Henry of 
Washington. William Thomson worked out the electro-static 
induction in submarine cables. The subject of the screening 
effect against induction, due to sheets of different metals, was 
worked out mathematically by Horace Lamb and also by 
Charles Niven. W. Weber's chief researches were on electro- 
dynamics. Helmholtz in 1851 gave the mathematical theory 
of the course of induced currents in various cases. Gustav 
Robert Kirchhoff ^ (1824-1887) investigated the distribution of 
a current over a flat conductor, and also the strength of current 
in each branch of a network of linear conductors. 
The entire subject of electro-magnetism was revolutionised 



APPLIED MATHEMATICS. 897 

by James Clerk Maxwell (1831-1879). He was born near 
Edinburgh, entered the University of Edinburgh, and became 
a pupil of Kelland and Forbes. In 1850 he went to Trinity 
College, Cambridge, and came out Second Wrangler, E. Routh 
being Senior Wrangler. Maxwell then became lecturer at 
Cambridge, in 1856 professor at Aberdeen, and in 1860 
professor at King's College, London. In 1865 he retired to 
private life until 1871, when he became professor of physics 
at Cambridge. Maxwell not only translated into mathematical 
language the experimental results of Faraday, but established 
the electro-magnetic theory of light, since verified experimen- 
tally by Hertz. His first researches thereon were published 
in 1864. In 1871 appeared his great Treatise on Electricity 
and Magnetism. He constructed the electro-magnetic theory 
from general equations, which are established upon purely 
dynamical principles, and which determine the state of the 
electric field. It is a mathematical discussion of the stresses 
and strains in a dielectric medium subjected to electro-magnetic 
forces. The electro-magnetic theory has received developments 
from Lord Rayleigh, J. J. Thomson, H. A. Rowland, R. T. 
Glazebrook, H. Helmholtz, L. Boltzmann, O. Heaviside, J. H. 
Poynting, and others. Hermann von Helmholtz turned his 
attention to this part of the subject in 1871. He was born 
in 1821 at Potsdam, studied at the University of Berlin, and 
published in 1847 his pamphlet Ueber die ErhaUung der Kraft, 
He became teacher of anatomy in the Academy of Art in 
Berlin. He was elected professor of physiology at Kdnigs- 
berg in 1849, at Bonn in 1855, at Heidelberg in 1858. It was 
at Heidelberg that he produced his work on Tonempfindung. 
In 1871 he accepted the chair of physics at the University of 
Berlin. From this time on he has been engaged chiefly on 
inquiries in electricity and hydrodynamics. Helmholtz aimed 
to determine in what direction experiments should be made to 



898 A HISTORY OF MATHEMATICS. 

decide between the theories of W. Weber, F. E. Neumann^ 
Eiemaim, and Clausius, who had attempted to explain electro- 
dynamic phenomena by the assumption of forces acting at a dis- 
tance between two portions of the hypothetical electrical fluid, — 
the intensity being dependent not only on the distance, but also 
on the velocity and acceleration, — and the theory of Faraday 
and Maxwell, which discarded action at a distance and assumed 
stresses and strains in the dielectric. His experiments favoured 
the British theory. He wrote on abnormal dispersion, and 
created analogies between electro-dynamics and hydrody- 
namics. Lord Kayleigh compared electro-magnetic problems 
with their mechanical analogues, gave a dynamical theory of 
diffraction, and applied Laplace's coefficients to the theory of 
radiation. Rowland made some emendations on Stokes' paper 
on diffraction and considered the propagation of an arbitrary 
electro-magnetic disturbance and spherical waves of light. 
Electro-magnetic induction has been investigated mathemati- 
cally by Oliver Heaviside, and he showed that in a cable it is 
an actual benefit. Heaviside and Poynting have reached 
remarkable mathematical results in their interpretation and 
development of Maxwell's theory. Most of Heaviside's papers 
have been published since 1882 ; they cover a wide field. 

One part of the theory of capillary attraction, left defective 
by Laplace, namely, the action of a solid upon a liquid, and 
the mutual action between two liquids, was made dynamically 
perfect by Gauss. He stated the rule for angles of contact 
between liquids and solids. A similar rule for liquids was 
established by Ernst Franz Neumann. Chief among recent 
workers on the mathematical theory of capillarity are Lord 
Rayleigh and E. Mathieu. 

The great principle of the conservation of energy was 
established by Robert Mayer (1814-1878), a physician in 
Heilbronn, and again independently by Colding of Copen- 



APPLIED MATHE&LA.TIG8. 899 

hagen, Joule, and Helmholtz. James Preacott Jonle (1818- 
1889) determined experimentally the mechanical equivalent 
of heat. Helmholtz in 1847 applied the conceptions of the 
transformation and conservation of energy to the various 
branches of physics, and thereby linked together many well- 
known phenomena. These labours led to the abandonment 
of the corpuscular theory of heat. The mathematical treat- 
ment of thermic problems was demanded by practical con- 
siderations. Thermodynamics grew out of the attempt to 
determine mathematically how much work can be gotten out 
of a steam engine. Sadi-Camot, an adherent of the corpuscular 
theory, gave the first impulse to this. The principle known 
by his name was published in 1824. Though the importance 
of his work was emphasised by B, P, E. Clapeyron, it did not 
meet with general recognition until it was brought forward 
by William Thomson. The latter pointed out the necessity 
of modifying Carnot's reasoning so as to bring it into accord 
with the new theory of heat. William Thomson showed in 
1848 that Carnot's principle led to the conception of an 
absolute scale of temperature. In 1849 he published '^an 
account of Carnot's theory of the motive power of heat, with 
numerical results deduced from Regnault's experiments.'' In 
February, 1850, Rudolph CUusius (1822-1888), then in Zlirich 
(afterwards professor in Bonn), communicated to the Berlin 
Academy a pa|)er on the same subject which contains thie 
Protean second law of thermodynamics. In the same month 
William John M. Rankine (1820-1872), professor of engineer- 
ing and mechanics at Glasgow, read before the Royal Society 
of Edinburgh a paper in which he declares the nature of 
heat to consist in the rotational motion of molecules, and 
arrives at some of the results reached previously by Clausius. 
He does not mention the second law of thermodynamics, but 
in a subsequent paper he declares that it could be derived 



400 A HISTORY OF MATHEMATICS. 

from equations contained in his first paper. His proof of 
the second law is not free from objections. In March, 1851, 
appeared a paper of William Thomson which contained a 
perfectly rigorous proof of the second law. He obtained it 
before he had seen the researches of Clausius. The state- 
ment of this law, as given by Clausius, has been much 
criticised, particularly by Rankine, Theodor Wand, P. G. 
Tait, and Tolver Preston. Repeated efforts to deduce it from 
general mechanical principles have remained fruitless. The 
science of thermodynamics was developed with great suc- 
cess by Thomson, Clausius, and Rankine. As early as 1852 
Thomson discovered the law of the dissipation of energy, 
deduced at a later period also by Clausius. The latter desig- 
nated the non-transformable energy by the name entropy, 
and then stated that the entropy of the universe tends 
toward a maximum. For entropy Rankine used the term 
thermodynamic function. Thermodynamic investigfations have 
been carried on also by G. Ad. Him of Colmar, and Helm- 
holtz (monocyclic and poly cyclic systems). Valuable graphic 
methods for the study of thermodynamic relations were de- 
vised in 1873-1878 by J. WiUard Gibbs of Yale College. 
Gibbs first gives an account of the advantages of using 
various pairs of the live fundamental thermodynamic quanti- 
ties for graphical representation, then discusses the entropy- 
temperature and entropy-volume diagrams, and the volume- 
energy-entropy surface (described in MaxwelPs Theoiry of 
Heat). Gibbs formulated the energy-entropy criterion of 
equilibrium and stability, and expressed it in a form appli- 
cable to complicated problems of dissociation. Important 
works on thermodynamics have been prepared by Clausius 
in 1875, by R. Riihlmann in 1875, and by Poincar^ in 1892. 

In the study of the law of dissipation of energy and the 
principle of least action, mathematics and metaphysics met on 



APPLIED MATHEMATICS. 401 

common ground. The doctrine of least action was lirst pro- 
pounded by Maupertius in 1744. Two years later he pro- 
claimed it to be a universal law of nature^ and the first 
scientific proof of the existence of (Jod. It was weakly sup- 
ported by him, violently attacked by Konig of Leipzig, and 
keenly defended by Euler. Lagrange's conception of the prin- 
ciple of least action became the mother of analytic mechanics, 
but his statement of it was inaccurate, as has been remarked 
by Josef Bertrand in the third edition of the M^canique Ana- 
lytique. The form of the principle of least action, as it now 
exists, was given by Hamilton, and was extended to electro- 
dynamics by F. E. Neumann, Clausius, Maxwell, and Helm- 
holtz. To subordinate the principle to all reversible processes, 
Helmholtz introduced into it the conception of the " kinetic 
potential." In this form the principle has universal validity. 
An offshoot of the mechanical theory of heat is the modem 
kinetic theory of gases, developed mathematically by C^usitLS, 
Maxwell, Lndmg Boltzmann of Munich, and others. The first 
suggestions of a kinetic theory of matter go back as far as the 
time of the Greeks. The earliest work to be mentioned here is 
that of Daniel Bernoulli, 1738. He attiibuted to gas-molecules 
great velocity, explained the pressure of a gas by molecular 
bombardment, and deduced Boyle's law as a consequence of 
his assumptions. Over a century later his ideas were taken 
up by Joule (in 1846), A. K. Kronig (in 1866), and Clausius 
(in 1857). Joule dropped his speculations on this subject 
when he began his experimental work on heat. Kr5nig 
explained by the kinetic theory the fact determined experi- 
mentally by Joule that the internal energy of a gas is not 
altered by expansion when no external work is done. Clausius 
took an important step in supposing that molecules may have 
rotary motion, and that atoms in a molecule may move rela- 
tively to each other. He assumed that the force acting 



402 A HISTORY OF MATHEMATICS. 

between molecules is a function of their distances, that tem- 
perature depends solely upon the kinetic energy of molecular 
motions, and that the number of molecules which at any 
moment are so near to each other that they perceptibly influ- 
ence each other is comparatively so small that it may be 
neglected. He calculated the average velocities of molecules, 
and explained evaporation. Objections to his theory, raised 
by Buy's-Ballot and by Jochmann, were satisfactorily answered 
by Clausius and Maxwell, except in one case where an addi- 
tional hypothesis had to be made. Maxwell proposed to him- 
self the problem to determine the average number of molecules, 
the velocities of which lie between given limits. His expres- 
sion therefor constitutes the important law of distribution of 
velocities named after him. By this law the distribution of 
molecules according to their velocities is determined by the 
same formula (given in the theory of probability) as the dis- 
tribution of empirical observations according to the magnitude 
of their errors. The average molecular velocity as deduced 
by Maxwell differs from that of Clausius by a constant factor. 
MaxwelFs first deduction of this average from his law of dis- 
tribution was not rigorous. A sound derivation was given by 
O. E. Meyer in 1866. Maxwell predicted that so long as 
Boyle's law is true, the coefficient of viscosity and the coeffi- 
cient of thermal conductivity remain independent of the press- 
ure. His deduction that the coefficient of viscosity should' 
be proportional to the square root of the absolute temperature 
appeared to be at variance with results obtained from pendu- 
lum experiments. This induced him to alter the ver}' foun- 
dation of his kinetic theory of gases by assuming between 
the molecules a repelling force varying inversely as the fifth 
power of their distances. The founders of the kinetic theory 
had assumed the molecules of a gas to be hard elastic spheres ; 
but Maxwell, in his second presentation of the theory in 1866, 



APPLIED MATHEMATICS. 403 

went on the assumption that the molecules behave like cen- 
tres of forces. He demonstrated anew the law of distribution 
of velocities ; but the proof had a flaw in argument^ pointed 
out by Boltzmann, and recognised by Maxwell, who adopted 
a somewhat different form of the distributive function in a 
paper of 1879, intended to explain mathematically the effects 
observed in Crookes' radiometer. Boltzmann gave a rigorous 
general proof of Maxwell's law of the distribution of velocities. 

None of the fundamental assumptions in the kinetic theory 
of gases leads by the laws of probability to results in very 
close agreement with observation. Boltzmann tried to estab- 
lish kinetic theories of gases by assuming the forces between 
molecules to act according to different laws from those pre- 
viously assumed. Clausius, Maxwell, and their predecessors 
took the mutual action of molecules in collision as repulsive, 
but Boltzmann assumed that they may be attractive. Ex- 
periment of Joule and Lord Kelvin seem to support the latter 
assumption. 

Among the latest researches on the kinetic theory is Lord 
Kelvin's disproof of a general theorem of Maxwell and Boltz- 
mann, asserting that the average kinetic energy of two given 
portions of a system must be in the ratio of the number of 
degrees of freedom of those portions. 



! 



t 



i.i 

.1 ' 



I 



INDEX. 



AbacUts, 126. 

Abacus, 8, 13, 63, 79, 82, 119, 122, 126, 

129. 
Abbatt, 334. 
Abel, 317, 348; ref. to, 146, 279, 291, 

312, 328, 336, 337, 360, 353, 371. 
Abelian functions, 292, 312, 328, 346, 

348, 349, 362, 366-367, 369. 
Abelian integrals, 360, 379. 
Abel's theorem, 362. 
Absolute geometry, 301. 
Absolutely convergent series, 336, 337, 

338. 
Abul Oud, 111 ; ref. to, 113. 
Abnl Hasan, 116. 
Abul Wefa, 110; ref. to, 112, 113. 
Achilles and tortoise, paradox of, 27. 
Acoustics, 262, 270, 278, 386. 
Action, least, 263, 366, 401; varying, 

292, 318, 379. 
Adams, 376 ; ref. to, 214. 
Addition theorem of elliptic integrals, 

262, 360, 396. 
Adrain, 276. 
.£quipollences, 322. 
Agnesi, 200. 
Agrimensores, 80. 

Ahmes, 10-16; ref. to, 17, 18, 63, 74. 
Airy, 376; ref . to, 383. 
Al Battani, 109; ref. to, 110, 126. 
Albertus Magnus, I'M. 
Albiruni, 111 ; ref. to, 102, 104. 
Alcnln, 119. 

Alembert, D'. See D*Alembert. 
Alexandrian School (first) , 34-^ ; (sec- 
ond ),.'VM>2. 
Alfonso's tables, 127. 



Algebra: Beginnings in Egypt, 15 
early Greek, 73; Diophantus, 74-77 
Hindoo, 93-96 ; Arabic, 107, 111, 115 
Biiddle Ages, 133, 136; Renaissance, 
140, 142-160, 162; seventeenth cen- 
tury, 166, 187, 192; Lagrange, 267; 
Peacock, 284; recent, 315-331; ori- 
gin of terms, 107, 116. See Nota- 
tion. 

Algebraic functions, 346; integrals, 
377. 

Algorithm, origin of term, 106 ; Mid- 
dle Ages, 126, 129. 

Al Haitam, 116; ref. to, 112. 

Al Hayyami, 112; ref. to, 113. 

Al Hazin, 112. 

Al Hogendi, 111. 

Al Karhi, 111, 113. 

Al Kaschi, 114. 

AlKuhi, 111; ref . to, 112. 

AlWgret, 377. 

Allman, IX., 36. 

Al MadshritI, 116. 

Almagest, 66^68; ref. to, 105, 109, 127, 
134, 136, 140. 

Al Mahani, 112. 

Alphonso's tables, 127. 

Al Sagani, 111. 

Alternate numbers, 322. 

Amp<^re, 394 ; ref. to, 361. 

Amyclas, 33. 

Analysis (in synthetic geometry), .'V), 
.'{9; Descartes', 186; modem, 331- 
3M. 

Analysis situs, 226, 316. 

Analytic geometry, 185-189, 191, 193, 
240, 287, 307-315. 



405 



406 



INDEX. 



Analytical Society (in Cambridge), 

283. 
Anazagoras, 18 ; ref . to, 28. 
Anaximander, 18. 
Anaximenes, 18. 
Angeli, 185. 
Anger, 375. 

Anharmonic ratio, 178, 291, 297, 306. 
Anthology, Palatine, 73, 120. 
Antiphon, 26; ref. to, 27. 
Apices of Boethins, 82; rel. to, 63, 

103, 119, 126, 129. 
Apollonian Problem, 50, 154, 188. 
Apollonios, 45-50; ref. to, 35, 37,40, 

54, 61, 66, 78, 105, 108, 115, 140, 153, 

154. 
Appel, 346. 
Applied mathematics, 373-403. ^ee 

Astronomy, Mechanics. 
Arabic manuscripts, 124-128. 
Arabic numerals and notation, 3, 73, 

87, 102, 112, 127-129, 159. 
Arabs, 100-117. 
Arago, XI., 332, 392. 
Arbogaste, 260. 
Archimedes, 40-45; ref. to, 2, 35, 37, 

39, 45, 47, 49, 60, 54, 61, 65, 73, 78, 90, 

105, 108, 140, 144, 169, 173, 182. 
Archytas, 23 ; ref. to, 29, 31, 32, 43. 
Areas, conservation of, 253. 
Arenarius, 65. 
Argand, 317 ; ref. to, 264. 
Aristaeus, 34 ; ref. to, 46. 
Aristotle, 34; ref. to, 9, 17, 27, 43, 61, 

68,125. 
Arithmetic: Pythagoreans, 20,67-70; 

Platonists, 29; Euclid, 38,70; Greek, 

63-77; Hindoo, 90-92; Arabic, 106; 

Middle Ages, 119, 122, 123, 126, 130, 

133, 134 ; Renaissance, 150, 151, 15^ 

161. See Numbers, Notation. 
Arithmetical machine, 220, 284. 
Arithmetical triangle, 196. 
Arraeraante, 313. 
Arneth, X. 
Aronhold, 327. 

Aryabhatta, 86; ref. to, 88, 91, 98. 
Aschieri, 305, 306. 
Assumption, tentative, 75, 92. See 

Regula falsa. 
Astrology, 155. 



Astronomy: Babylonian. 8 ; Egyptian, 
10 ; Greek, 18, 24, 32, 39, 51, 56 ; Hin- 
doo, 86; Arabic, 100, 101, 105, 115; 
Biiddle Ages, 127 ; Newton, 212-216 ; 
more recent researches, 253, 257, 
262, 271-274, 366, 373-377. See Me- 
chanics. 

Athelard of Bath, 125; rel. to, 135. 

Athemeus, 32. 

Atomic theory, 382. 

Attains, 46. 

Attraction, 277. ^ee Grayitation, El- 
lipsoid. 

August, 296. 

Ausdehnungslehre, 320, 321, 378. 

Axioms (of geometry), 30, 37, 38, 281, 
300, 315. 

Babbage, 283, 366. 

Babylonians, 5-9 ; ref. to, 19, 51. 

Bachet de Mdziriac. See M€ziriac. 

Bachmann, 371; ref. to, 365. 

Bacon, R., 134. 

Baker, Th., 113. 

Ball, Sir R. S., 378. 

Ball, W. W. R., X, 217. 

Ballistic curve, 279. 

Baltzer, R., 314; ref. to, 302, 325. 

Barbier, 341. 

Barrow, 198; ref. to, 173, 202, 203, 

221,227. 
Basset, 380, 382. 
Battaglini, 306. 
Bauer, XII. 
Baumgart, XI. 
Bayes, 340. 
Beaumont, XI. 

Beaune, De. See De Beaune. 
Bade, the Venerable, 118. 
Beer, 392. 
Beha Eddin, 114. 

Bellavitis. 322; ref. to, 300, 304, 317. 
Beltrami, 30i, 305; ref. to, 315. 
Ben Junus, 115. 
Berkeley, 236. 
Bernelinus, 122. 
Bernoulli, Daniel, 238; ref. to, 255, 

262, 386, 401. 
Bernoulli, Nicolaus (born 1695), 238. 
Bernoulli, Nicolaus (bom 1687), 239, 

251,269. 



INDEX. 



407 



Bemonlli, James (bom 1654) , 237, 238 ; 

ref . to, 182, 226, 229, 251. 
Bernoulli, James (born 1758), 239, 356, 

387. 
Bernoulli, John (bom 1667), 238; ref. 

to, 226, 229, 232, 234, 237, 243, 250, 

251,356. 
Bernoulli, John (born 1710), 239. 
Bernoulli, John (bora 1744), 239. 
Bernoullis, genealogical table of, 236. 
Bemonlli's theorem, 237. 
Bertini, 305. 
Bertrand, 337, 340, 342, 377, 379, 380, 

401. 
Bessel, 373-375; ref. to, 303, 309, 351. 
Bessel's functions, 374. 
Bessy, 181. 
Beta function, 249. 
Betti, 353. 
Beyer, 160. 

Bezout, 260; ref. to, 250, 264. 
Bezout's method of elimination, 260, 

331. 
Bhaskara, 87 ; ref. to, 92-96, 97, 152. 
Bianchi, 328. 
Billinfs^ley, 138. 
Binet, 324, 387. 
Binomial formula, 195, 196, 202, 251, 

at8. 
Biot, 275, 288, 393. 
Biquadratic equation, 112, 146, 149. 
Biquadratic residues, IVJti. 
Biquaternions, 378. 
Bjerknes, C. A., XIII., 357, 383. 
Bobillier, 308. 
IkVher, XIV. 
Bode, 341. 
Boethiufl, 81 ; ref. to, 63, 72, 103, 118, 

121, KH, 135. 
Rois-R4>yniond,P.du,XIII.,337-339,362. 
Boltzmann, 3^17, 403. 
Bolyai, Johann, 302; ref. to, 291. 
Bolyai, Wolfgang, 301, 302; ref. to, 

291, ^A. 
Bolza, .'{50. 

Bombelli. 146; ref. to, 152. 
Bonnet, O., 314; ref. to, 337, .%42. 
Boole, M3\ ref. to, 291, 325. .'{40, 341, 

346. 
Booth, 311. 
Borchardt, :Wi. 



Boaniakowsky, 365. 

Bouquet, 344; ref. to, 346, 354. 

Bour, 341, 377. 

Boussinesq, 383, 391, 393. 

Bowditch, 275, 323. 

Boyle's law, 401. 

Brachistochrone (line of swiftest de- 
scent), 234, 238. 

Bradwardine, 135 ; ref. to, 141. 

Brahe, Tycho, 110, 139, 168. 

Brahmagupta, 86; ref. to, 92, 95, 96, 
102. 

Bredon, 135. 

Bretschneider, IX, 97, 320. 

Brianchion, 178, 288, 289. 

Briggs, 163. 

BriU, A., 297, 311, 356. 

Brill, L., 307. 

Bring, 328. 

Brioschi, 327 ; ref. to, 326, 330, 334, 350, 
363,379. 

Briot, 344; ref. to, 346, 364. 

Brouncker, 197. 

Brano, Fak de, 327. 

Brans, 377. 

Bryson of Heraclea, 27. 

Buchheim, 378; ref. to, 306. 

Buckley, 159. 

Budan, 282. 

Buddha, 89. 

Buffon, 340. 

Bungus, 155. 

Burgi, 160; ref. to, 166. 

Burkhardt, H., XII., 328. 

Burkhardt, J. K., 275. 

Burmester, 300. 

Busche, 365. 

Buteo, 154. 

Buy's-BaUot, 402. 

Byrgius. See BiirgL 

Cesar, Julius, 81. 

Calculating machines, 220, 284. 

Calculation, origin of word, 79. 

Calculus. ISee Differential Calculus. 

Calculus of operations, 2^)2 ; of varia- 
tions, 247, 249, 261, 265, 296, 328, 
Xa-Qii^, 3»>, 366. 

Calendar, 9, 81, 141, 154, 271. 

Callisthenes, 9. 

Canon paachalis, 79. 



408 



INDEX. 



Cantor, G., 339, 362, 372. 

Cantor, M., IX., X., 112. 

Capelli, 330. 

Capillarity, 278, 366, 388, 308. 

Caporali, 313. 

Cardan, 144; ref. to, 149, 162, 105, 156, 

169. 
Carll, 334. 
Camot, Lazare, 288, 289; ref. to, 66, 

236, 2a3. 
Camot, Sadi, 399. 
Casey, 313. 
Cassini, D., 257. 
CassiodoriuB, 83, 118. 
Casting out the 9'b, 91, 106. 
Catalan, E., 325. 
Cataldi, 159. 
Catenary, 191, 231, 237. 
Cattle-problem, 73. 
Cauchy, 331-333; ref. to, 243, 247, 264, 

322, 324, 328, 330, 335, 338, 339, 341. 

342, 345, 318, 349, 350, 353, 366, 362, 

368, 383, 387, 388, 390, 393. 
Caustics, 238, 241. 

Cavalieri, 170; ref. to, 167, 193, 221. 
Cavendish, 394. 
Cayley, 325, 326; ref. to, XII., XIV., 

291, 296, 297, 306, 308, 311, 313, 319, 

324, 330, -^6, 354, 355. 
Centre of gravity, 177, 191 ; of oscilla- 
tion, 191, 243. 
Centres of osculation, 49. 
Centrifugal force, 183, 192, 214. 
Ceulen, van. See Ludolph. 
Ceva, 290. 
Chapman, 324. 

Characteristics, method of, 297. 
Chasles, 296-298; ref. to, X., 39, 47, 

49, 52, 172, 289, 294, 306, 311, 313, 377. 
Chauvenet, 377. 
Chess, 92. 
Cheyne, 206. 
Chinese, 19. 
Chladni's figures, 386. 
Chree, 382, 391. 
Christoffel, 325, 327. 
Circle, 10, 24-28, 31, 41, 52, 154, 194 ; 

degrees of, 7, 271 ; division of, 329, 

365. 
Circle-sqiiarers, 2, 19, 1<K), 316. 
Cissoid, 50, 191. 



Clairaut, 25&-258; ref. to, 244, 262, 265, 
262. 

Clapeyron, 399. 

Clarke, 341. 

Clausius, 399; ref. to, 390, 396, 400- 
402. 

Clavius, 155 ; ref. to, 164. 

Cleb8ch,312.313; ret to, XII., 296, 309, 
315, 322, 327, 328, 333, 341, 342, 358, 
381, 383, 390-392. 

Clifford, 305. 306 ; ref. to, 297, 319, 324, 
368, 378, 396. 

Cockle, 316. 

Colbum, Z., 180. 

Colding, 998. 

Cole, 330. 

Colebrooke, 87. 

Colla, 143, 145. 

Collins, 203, 223, 227, 228, 230. 232. 

Colson, 201. 

Combinatorial School, 247, 336. 

Commandinus, 163. 

Commercium epistolicum, 206, 232. 

Complex quantities, 292, 317. See Im- 
aginaries. 

Complex of lines, 309. 

Computus, 118, 119. 

Comte, X. 

Concentric spheres of Eudoxus, 32. 

Conchoid, 50. 

Condensation of singularities, 362. 

Conform representation of surfaces, 
361. 

Congruencies, theory of, 365. 

Congruency of lines, 309. 

Conic sections, Greek, 32, 34, 40, 41, 
45-49, 55; Arabs, 101, 112; Renais- 
sance, 153; Kepler, 168; more re- 
cent researches, 176-178, 192. See 
Geometry. 

Conon, 40; ref. to, 42. 

Conservation of areas, 253 ; of energy, 
397, 398 ; of vis viva, 192. 

Continued fractions, 159, 197. 252, 270. 

Continuity, 169, 193, 226, 293, 333, 359, 
372. 

Contracted vein, 384. 

Contravariants, 326. 

Convergence of series, 334-339. 

Co-ordinates, 185, 294, 308. 314, 379; 
first use of term, 226. 



INDEX. 



409 



Copemican System, 139. 

Copernicus, 56, 139. 

Correspondence, principle of, 293, 297. 

Cosine, 1<>5. 

Coss, term for algebra, 152. 

Cotangent, 141, 166. 

Cotes, 242 ; ref . to, 243. 

Coulomb, 394. 

Cournot, 340. 

Cousinery, 299. 

Covariants, 327, 354, 369. 

Cox, 306. 

Craig, 306, 3i7, 355, 382. 

Craige, 226. 

Cramer, 217. 

Crelle, 348; ref. to, 349. . 

Crelle's Journal, 295. 

Cremona, 299; ref. to, 291, 294-296, 
300, 313. 

Cridhara, 87. 

Criteria of convergence, 334-339. 

Crofton, 341. 

Crozet, 288. 

Ctesibius, 52. 

Cube, duplication of. See Duplication 
of tbe cube. 

Cube numbers, 72, 111, 180. 

Cubic curves, 217, 257, 297. 

Cubic equations, 112, 113, 142-145, 149, 
152,153. £«e Algebra. 

Cubic residues, 366. 

Culmann, 299, 300. 

Cnrtze, M., 299. 

Curvature, measure of, 314. 

Curve of swiftest descent, 234, 238. 

Curves, osculating, 226 ; quadrature of, 
42, 49, 177, 190, 193, 202, 220; theory 
of, 226, 240, 242, 243, 292, 321. See. 
Cubic curves, RectiiScation, Geom- 
etry, Conic sections. 

Cusanus, 154. 

Cyclic method, 96, 97. 

Cycloid, 171, 173, 176, 187, 190, 191, 
225, 234, 240. 

Cyzicenus, 33. 

Czuber, 340. 

D*Alembert, 254-256 ; ref. to, 254, 258, 

262. 2«». 268-270, 386. 
D'Alembert*8 principle, 254. 
Damascius, 61 ; ref. to, 38, 104. 



Darbouz, XIII., 313, 343, 346, 361, 362. 

Darwin, 376; ref. to, 384, 391. 

Data (£ucUd's),39. 

Davis, E.W., 306. 

Davis, W. M., 385. 

De Baune, 189; ref. to, 185, 223, 225. 

Dee, 138. 

Decimal fractions, 159-161. 

Decimal point, 161. 

Dedekind, 371 ; ref. to, 357, 362, 372. 

Deficiency of curves, 312. 

Definite integrals, 169, 334, 339, 341, 
351,362. 

Deinostratns. See Dinostratus. 

De Lahire, 285, 290. 

Delambre, 366. 

Delaunay, 376 ; ref. to, 333, 334. 

Delian problem. See Duplication of 
the cube. 

Del Pezzo, 305. 

Democritus, 28 ; ref. to, 16. 

De Moivre, 240. 

De Morgan, 316 ; ref. to, X., XL, 1, 2, 
70, 96, 161, 205, 229, 233, 260, 277, 
285, 291, 333, 337, 340, 366. 

De Paolis, 306. 

Derivatives, method of, 269. 

Desargues, 177 ; ref. to, 174, 184, 240, 
285,290. 

Desboves, 379. 

Descartes, 183-189; ref. to, 4, 48, 60, 
113, 167, 173, 174, 189, 191, 192, 216, 
220, 223, 240, 317 ; rule of signs, 187, 
193. 

Descriptive geometry, 286-288, 300. 

Determinants, 226, 265, 278, 313, 324, 
325, 331, 363. 

Devanagari-numerals, 103. 

Dialytic method of elimination, 330. 

Differences, finite. See Finite dif- 
ferences. 

Differential calculus, 200, 221-227, 236- 
242 {see Bemoullis, Euler, La- 
grange, Ijiplace, etc.) ; controversy 
between Newton and Leibniz, 227- 
233; alleged invention by Pascal, 
174 ; philosophy of, 236, 256, 259, 268, 
289. 333. 

Differential equations, 239, 252, 266, 
278, 314, 318, 321, 333, 341-347. 

Differential invariants, 327. 



410 



U7DEX. 



Dingeldey, 315. 

Dini, 337 ; ref. to, 362. 

Dinostratus, 32 ; ref. to, 25. 

Diocles, 50. 

Diodorus, 10, 40, 58. 

Diogeues Laertius, 17, 32. 

Dionysodorus, 54. 

Diophantos, 74-77 ; ref. to, 55, 61, 86, 

93, 95, 96, 105, 106, 107, 110, 111, 179, 

372. 
Directrix, 49, 60. 
Dirichlet, 367-369; ref. to, Xm., 179, 

291, 334, 338, 339, 348, 366, 357, 369, 

362, 371, 396. 
Dissipation of energy, 400. 
Divergent parabolas, 217, 267. 
Divergent series, 255, 337. 
Division of the circle, 7, 271, 329, 

365. 
Diwani-nnmenUs, 102. 
Donkin, 379. 
Dositheos, 40. 
Dostor, 325. 
Dov^, 384. 
D'Ovidio, 306. 
Dronke, XII. 
Duality, 290, 297, 308. 
Duhamel, 333, 388. 
Dtihring, £., X. 
Duillier, 230. 
Duodecimals, 124, 126. 
Dupin, 288, 289; ref. to, 300, 314. 
Duplication of the cube, 23-25, 31, 32, 

45, 50, 153. 
Dur^ge, .^54 ; ref. to, 309, 315. 
Diirer, A., 156. 
Dusing, 340. 
Dyck, 315. See Groaps. 
Dynamics, 318, 378-381. 
Dziobek, XIU., 377. 

Eamshaw, 383. 

Earth, figure of, 257,292; rigidity of, 

391 ; size of, 214, 215. 
Eddy, 300. 
Edfu, 12, 53. 
Edge worth, 340. 
Egyptians, 9-lf>, 19. 
Eisenlohr, 3Xr 
Eisenstein, 369; ref. to, 354, 357, 365, 

370, 371. 



Elastic cnrve, 237. 
Elasticity, 278, 387-392. 
Electricity, 394-^398. 
Electro-magnetic theory of light, 991. 
Elements (Euclid's), 36-^, 104, 114, 

125, 127, 128, 133, 135, 136. 138. See 

Eaclid. 
Elimination, 250,308,310,330.331. ^ee 

Equations. 
Elizabeth, Princess, 188. 
Ellipsoid (attraction of), 215, 277, 280, 

285, 298, 366, 378, 379; motion of. 

Elliptic co-ordinates, 379. 

Elliptic functions, 241, 279, 280, 296, 

329, 345, 346, 347-354, 363, 367, 370. 
Elliptic geometry. See Non-Eaclidean 

geometry. 
Elliptic integrals, 247, 252, 328, 349, 

350. 
Ely, 372. 
Encke, 366. 

Energy, conservation of, 397, 398. 
Enestrom, XI. 
Enneper, 353; ref. to, XIII. 
Entropy, 400. 

Enumerative geometry, 297. 
Epicycles, 51. 
Epping, IX., 9. 
Equations, solution of, 15, 149, 153, 

186, 250, 260, 263, 277, 348; theory 

of, 75, 166, 189, 193, 216, 240, 241, 

260, 32^331; numerical, 147, 264, 

282. See Cubic equations. Algebra, 

Theory of numbers. 
Eratosthenes, 44 ; ref. to, 25, 35, 40, 71. 
Errors, theory of. See Least squares. 
Espy, 384. 

Ether, luminiferons, 393. 
Euclid, 35-40, 70, 71; ref. to, 17, 21. 

22, 26, 30, 31, a3, 34, 42, 46, 50, 63. 

57, 58, 61, 72, 73, 78, 81, 97, 104, 108, 

114, 125, 127, 136, 138, 144, 162, 281, 

303. 
Euclidean space. See Non-Euclidean 

geometry. 
Eudemian Summary, 17, 21, 30, 32, 33, 

35. 
Eudemus, 17, 22, 45, 46, 69. 
Eudoxus, 32, 33; ref. to, 16, 28, 31, 32, 

35, 36, 51. 



INDEX. 



411 



Euler, 24^254; rof. to, 77,96, 179, 180, 
2:», 241, 246, 260, 258, 259, 261, 2<)2, 
2G4, 265, 267, 268, 273, 278, 279, 280, 
287, 314, 317, 334, 3(H, 365, 367, 374, 
:J86, :«7, 401. 

Rulerian integrals, 280. 

Kiitocias, 61 ; ref. to, 45, 46, 54, 65. 

E volutes, 49, 191. 

Exhaustion, method of, 26, 28, 33, 36, 
42, 169. 

Exponents, 134, 152, 160, 162, 187, 202, 
241. 

Factor-tobies, 368. 

Fagnano, 241. 

Fahri des Al Karhi, 111. 

Falsa positio, 92, 147. 

Faraday, 396. 

Favaro, XII. 

Faye, 377. 

Format, 173, 170-182; ref. to, 172, 173, 

177, 198, 252, 264, 265, 367. 
Fermat's theorem, 180, 252. 
Ferrari. 145; ref. to, 144, 2154. 
Ferrel, ;«4 ; ref. to, 376. 
Ferro, Scipio, 142. 
Fibonacci. See Leonardo of Pisa. 
Fiedler, :iOO, 312, 328. 
FijTure of the earth, 257, 292. 
Fiiiffius, 159. 
Fine, XII. 

Finder-reckoning, 63, 118. 
Finite differences, 240, 242, 251, 270, 

27H, *43. 
Fink, XII. 
Fitzgerald, 394. 
Flachenabbildung, 313. 
Fl:iniHtee<l, 218. 
Floridas, 142, 144. 
Fiexure, theory of, 389. 
Fluents, 205, 206. 
Fluxional controversy, 227-233. 
Fluxions, 200, 202-213, 333. 
Focus, 49, 60, 169, 170. 
Fontoine, 252, 2&4. 
Forbes, 397. 

Force-fanction, 396. See Potential. 
Forsyth, XII., 327, 344, 362. 
Four-point problem, 341. 
Fourier, 281-284; ref. to, 174, 255, 351, 

366,367. 



Fourier's series, 283, 338, 339, 367, 
386. 

Fourier's theorem, 282. 

Fractions, Babylonian, 7; Egyptian, 
13; Greek, 26, 64, 65; Roman, 78; 
Hindoo, 94; Middle Ages, 120, 124; 
decimal, 159, 160; sexagesimal, 7, 
57, 65, 67, 126; duodecimal, 124, 126; 
continued, 159, 197, 252, 270. See 
Arithmetic. 

Franklin, :J27, 372. 

Frantz, XIII. 

Fresnel, 392. 

Fresnel's wav&«arface, 209, 314. 

Fr^zier, 286. 

Fricke, 354. 

Friction, theory of, 382. 

Frobenius, 325, 344, 345. 

Frost, 315. 

Froude, :M1, 384. 

Fucha, im ; ref. to, 344, 345. 

Fuchsian functions, 345, 360. 

Fuclisian groups, 'MR. 

Functions, definition of, 366; theory 
of, 268, 269, 345, 35&-362 ; arbitrary, 
2<)2, 283. See Elliptic functions, 
Abelian functions, Hyperelliptic 
functions, Theto functions, Beto 
function. Gamma function. Omega 
function, Sigma function, Beasers 
function. Potential. 

Funicular polygons, 299. 

Gabir ben Aflah, 115; ref. to, 127. 

Galileo, 182; ref. to, 43, 139, 161, 168, 
170, 171, 188. 

Galois, 329. 

Gamma function, 249. 

Garbieri, 324. 

Qtuses, Kinetic theory of, 401-403. 

Gauss, :«3-367 ; ref. to, 77, 158, 247, 
248, 251, 264, 276, 291, 291, 302, 308, 
304, 313, 314, 315, 317, :«0, 324. 326, 
2:)9, .'t30. 33:i. 335, 343, 348, 360, 351, 

:tf7. :«J2. 373, :m. 

Gauss' Analogies, .'S66. 
(teber. See Gabir ben Aflah. 
Gel»er*8 thf'orem, 116. 
Geminus, 53; ref. to, 46, 60, 67. 
(Sellibrand. KA. 
Genocchi, 366. 



412 



INDEX. 



Geodesies, 249, 379. 

Geodesy, 366. 

Geometry, Babylonian, 8; Egsrptian, 
10-13; Greek, 17-62, 69; Hindoo, 97, 
98; Roman, 80; Arabic, 10^, 108, 
110, 113, 114; Middle Ages, 121, 125, 
127, 128, 130, 131 ; Renaissance, 138, 
153, 154, 158, 167 ; analytic, 186-189, 
191, 193, 287, 307-315; modern syn- 
thetic, 240, 285-290, 29^-307; de- 
scriptive, 286-288, 300. See Curves, 
Surfaces, Curvature, Quadrature, 
Rectification, Circle. 

Gerard of Cremona, 126. 

Gerbert, 120-124. 

Gergonne, 297 ; ref . to, 178, 290. 

Gerhardt, XI., 227, 230, 233. 

Gerling, 366. 

Germain, Sophie, 387 ; ref. to, 386. 

Grerman Magnetic Union, 366. 

G^rstner, 389. 

Gibbs, 400; ref. to, XII., 319. 

Giovanni Campano, 127. 

Girard, 166; ref. to, 127, 161. 

Glaisher, 372; ref. to, XIH., 326, 328, 
374,368. 

Glazebrook, 397; ref. to, XIV. 

Gk>bar numerals, 82, 103. 

Godfrey, 218. 

Grolden section, 33. 

Gopel, 355. 

Gordan, 312, 327, 330. 

Gournerie, 300, 311. 

Goursat. 343; ref. to, 360. 

Gow, IX., 35. 

Graham, XII. 

Grammateus, 151. 

Grandi, 251. 

Graphical statics, 292, 299. 

Grassmann, 320-321 ; ref. to, 29i, 304, 
317, 318, 378. 

Gravitation, theory of, 213, 258, 271, 
275. 

Greeks, 16-77. 

Green, 395; ref. to, 358, 383, 388, 390, 
393, .3f<5. 

Greenhill, 354. 382. 

Gregorian Calendar, 154. 

Gregory, David F., 215, 284, 315. 

Gregory, James, 228, 243. 

Gromatici, 80. 



Groups, theory of, 328-330, 344-346. 
Papers by W. Dyck (Math. Ann., 20 
and 22) and by O. Holder (Math. 
Ann., 34) should have been men- 
tioned on p. 330. 

Grunert, 314; ref. to, 320. 

Gua, de, 240. 

Gubar-numerals, 82, 103. 

Gudermann, 363. 

Guldin, 167 ; ref. to,.69. 171. 

GuldinuB. See Guldin. 

Gunter, E., 165. 

Gunther, S., IX., X., XI.» 325. 

Giitzlaff. 353. 

Haan, 334. 

Haas, XII. 

Hachette, 288. 300. 

Hadamard, 368. 

Hadley. 218. 

Hagen, 276. 

Halifax, 134; ref. to, 136. 

Halley, 46, 213, 214, 261. 

Halley's Comet, 258, 374. 

Halphen, 311; ref. to, 297, 315, 327. 

a44, 345, 354. 
Halsted, X., 303. 
Hamilton, W., 184,316. 
Hamilton, W. R., 318, 319; ref. to, 

260, 201, 292, 314, 310, 317, 321, 324, 

328, ;U1, 378, .371), 393. 401. 
Hamilton's numbers, 329. 
Hammond, J., 327. 
Hankel, 322 ; ref. to, IX., X., 28, 93, 

96. 285, .'^5, 339, 362. 
Hann, 385. 
Hansen, 375. 
Hanus, 325. 
Hardy. 174. 
Harkuess, 362. 
Harmonics, 55. 
Haroun-al-Raschid, 104. 
Harrington, 377. 
Harriot, 166; ref. to, 147, 152, 162, 187, 

192. 
Hathaway, XI. 
Heat, theory of, 399-401. 
Heatli, 306. 

Heaviside, 319, 397, 398. 
Hebrews, 19. 
Hegel, 373. 



INDEX. 



418 



Heine. 339; ref. to, 362. 372. 

Helen of geometers, 187. 

Helicon, 32. 

Heliotrope, 363. 

Helmholtz, 397, 398; ref. to, 301, 306, 

382, 385, 386, 393. 396, 400, 401. 
Henrici, XIU. 
Henry, 396. 
Heraclides, 46. 
Hermite, 363; ref. to. XUI.. 328. 330, 

343, 347, 360. 356, 362, 372. 
Hermotimus, 33. 
Herodianic signs, 63. 
Heron the Elder, 62 ; ref. to, 60, 64, 66, 

80, 96, 106, 131, 140. 
Herschel, J. F. W.. 386; ref. to, X., 

276, 283, 284, 366. 
Hesse, 309-311 ; ref. to, 295, 309, 312, 

326, 329, 330,- 333, 342. 377. 
Hessian, 296, 310, 327. 
Heuraet, 190. 

Hezagrammum mysticom, 178, 296. 
Hicks, 382, 3M. 
Hilbert. 327. 
Hill, 376. 

Hindoos, 84-100; ref. to, 3. 
Hipparchns, 61; ref. to, 64, 66. 
Hippasus, 22. 
Hippias of Elis, 25. 
Hippocrates of Chios, 25, 28, 30. 
Hippopede, 51. 
Him, 400. 

History of mathematics, its yaliie, 1-4. 
Hodgkinson, 389. 
Holder, O. See Groups. 
Holmboe, 336, 347. 350. 
Homogeneity, 293, 308. 
Homological figures, 178. 
Honein ben Ishak, 104. 
Hooke. 21.1. 
Hoppe, 306. 
Homer, 147, 330. 
Hospital, r, 239, 240. 
Houel. 319. 
Hovarezmi, 106; ref. to, 107. 110, 114, 

126, 127. 
Hndde, 189; ref. to. 203. 
Hurwitz, 358. 
Hossey, 377. 
Hnygens, 190-192; ref. to. 177, 182, 

188, 213, 214, 219, 234, 257, 302. 



Hyde. 321. 

Hydrodynamics. 239, 255, 380, 381-384. 
See Mechanics. 

Hydrostatics, 44, 265. ^ee Mechanics. 

Hypatia, 61 ; ref. to. 37. 

Hyperbolic geometry. See Non-Eucli- 
dean geometry. 

Hyperelliptic functions, 292, 328, 348, 
364,360. 

Hyperelliptic integrals, 352. 

Hypergeometric series, 335, 361. 

Hyperspace, 304, 306. 

Hypsicles. 61 ; ref. to, 7, 38. 71. 104. 

lamblichus, 72; ref. to. 10. 22. 69. 

Ibbetson, 392. 

Ideal numbers, 371. 

Ideler. 32. 

lehuda ben Mose Cohen. 127. 

Ignoration of co-ordinates, 380. 

Images, theory of. 381. 

Imaginary geometry, 301. 

Imaginary points, lines, etc, 298. 

Imaginary quantities, 146, 166, 241, 
287. 349. 363. 372. 

Imschenetzky, 342. 

Incommensurables, 36, 38, 70. See 
Irrationals. 

Indeterminate analysis. 95, 101, 111. 
See Theory of numbers. 

Indeterminate coefficients, 186. 

Indeterminate equations, 95. 101, 111. 
See Theory of numbers. 

Indian mathematics. See Hindoos. 

Indian numerals. See Arabic numer- 
als. 

Indices. See Exponents. 

Indivisibles. 170-173, 176, 198. 

Induction. 340. 

Infinite products, 349, 354. 

Infinite series. 197. 203. 208. 220, 247, 
260. 266. 259, 269, 283, 334^330. 348, 
349. 361. 363. 

Infinitesimal calculus. See Differen- 
tial calculus. 

Infinitesimals. 135. 169, 207, 208, 211. 

Infinity. 27. 136. 169. 178. 193. 269, 293, 
."tOI. 308; symbol for, 193. 

Insurance. 239. 340. 

Integral calculus, 171, 223, 348. 300, 
368, 371 ; origin of term, 237. 



414 



INDEX. 



Interpolation, 191. 

Invariant, 293, 310, 325, 328, 344, 354. 

Inverse probability, 340. 

Inverse tangents (problem of), 169, 189, 

220, 222, 223. 
Involution of points, 60, 177. 
Ionic School, 17-19. 
Irrationals, 22, 26, 69, 94, 107, 362,372. 

See Incommensurables. 
Irregular integrals, 344. 
Ishak ben Honein, 104. 
Isidorus of Seville, 118 ; ref . to, 61. 
Isochronous curve, 234. 
Isoperimetrical figures, 61, 237, 249, 

261. See Calculus of variations. 
Ivory, 285 ; ref. to, 276. 
Ivory's theorem, 285. 

Jacobi, 351-352; ref. to, 279, 291, 295, 
308, 309, 315, 324, 330, 333, 341, 347, 
349, 350, 353, 357, 365, 367, 370, 374, 
377, 378, 379, 381. 

Jellet, 334; ref. to, 381, 390. 

Jerrard, 328. 

Jets, 382, 386. 

Jevons, 340. 

Joachim. See Rhseticus. 

Jochmann, 402. 

John of Seville, 126, 159. 

Johnson, 347. 

Jordan, 329; ref. to, :i41, ai3, »4(>. 

Jordanus Nemorarius, 134. 

Joubert, 353. 

Joule, 399; ref. to, 401, 403. 

Julian calendar, 81. 

Juriu, 2;)0. 

Kaestner, 363; ref. to, 217. 

Kant, 274, 37G. 

Kauffinaiin. See Mercator, X. 

Keill,231,232, 235. 

Kellaud, :iH;3, 397. 

Kelvin, Ix)rd, 39r>-3<)G; ref. to, 283, .'tis, 
358, 381, 382, 383, :i8«, 391, 3<i;5, 314, 
3i>5, 31«>, 400, 403. See Thoinsou, W. 

Keinpe, 3'J<). 

Kepler, 1(W-170 ; ref. to, 139, ir»<), 158, 
KJl, 11)7, 171, 174, 202, 213, 203. 

Kepler's laws, 1G8, 213. 

Kerbedz, XIIl. 

Ketteler, 393. 



Killing, 306. 

Kinckhuysen, 204. 

Kinetic theory of gases, 401-403. 

Kirchlioff, 396; ref. to. 309, 381, 382, 

«>Ot5, tioo, o9«5, u9d. 
Klein, 343; ref. to, XH., 305, 306, 307, 

309, 313, 328, 330, 345, 346, 347, 354. 
Kleinian groups, 345. 
Kleinian functions, 360. 
Kohlrausch, 394. 
Kohn, 337. 
Konig, 401. 
Konigsberger, 353; ref. to, 344, 360, 

354,355. 
Kopcke, 382. 

Korkine, 372 ; ref. to, 341. 
Korndorfer, 313. 

Kowalevsky, 380 ; ref. to, 345, 352, 378. 
Krause, 355. 
Krazer, 355. 

Kronecker.329; ref. to, 328, 330, 359. 366. 
Kronig, 401. 
Kuhn, H., 317. 
Kuhn, J., 219. 
Kummer, 370, 371; ref. to, XIII., 179, 

314, 337, 338, 343, 355, 365. 

Lacroix, 284, 286, 287, 320. 

Laertius, 10. 

Lagrange, 260-270; ref. to, 4, 77, 174, 
179, 183, 244, 246, 247, 248, 254, 255, 
259, 273, 277, 27S, 279, 280, 2i)3, 2*16, 
304, 309, 313, 314, 325, 362, 363, 364, 
367. 378, 383, 386, 387, 401. 

Laguerre, 30<). 

Lahire, de, 240. 

Laisant, 319. 

La Louere, 177. 

Lamb, 378, 382, 383, 396. 

Lambert, 258-259 ; ref. to, 2, 2<0, 2XX\, 
313. 

Lame, 389 ; ref. to, 367, 389, 392. 

Lame''s functions, .'i89. 

Laiidon, 259; ref. to, 2r>8, 279. 

Laplace, 270-278; ref. to, 174, 215, 245, 
246, 256, 2(;3, 279, 285, 320, 336, 340, 
362, ;i(>3, 373, 375, 376, .'384, 386, 392, 
395, ;i98. 

I^place's coefficients, 277. 

Latitude, periodic changes in, 392. 

Latus rectum, 48. 



INDEX. 



415 



Laws of Laplace, 273. 

I^W8 of motion, 183, 188, 213. 

Least action, 253, 2(i6, 401. 

Least squares, 270, 281, 285, 363. 

Lebesgue, :J25, 333, 3G5. 

Legendre, 278-281 ; ref. to, 247, 252, 

259, m\, 27<>, 301, ^49, 350, 351, 353, 

365,307. 
Lcgendre's function, 280. 
Leibniz, 21<V-2:»; ref. to, 4, 158, 176, 

200, 208, 209, 210, 237, 241, 250, 251, 

252, 268, 315, 334, 366. 
Lemoine, 341. 
Lemonnier, 267. 
Lcodanias, 3.'$. 
Leon, 33. 

I^eonardo of Pisa, 128 ; ret to, 133, 137. 
Leslie, X. 

Le Verrier, 375 ; ref. to, 376. 
Levy, 300, 301. 
Lewis, 382. 
Lexis, 340. 
Leyden jar, 396. 

L'Hospital, 2:{9, 240; ref. to, 229, 234. 
Lie,:M6; ref. to, 311, 350. 
Light, theory of, 218, 390. 
Limits, method of, 212, 268. 
Lindeluf. 3:^. 

Lindemann, 315 ; ref. to, 2, 306, 366. 
Linear associative algebra, 323. 
Lintearia, 237. 
Liouville, 369; ref. to, 314, 366, 366, 

370, :m). 

Lipschitz, tVX} ; ref. to, 338, 376, 382. 

Listing, 315. 

Lloyd, 393. 

Lobatchewsky, 301 ; ref. to, 291, 303. 

Local probability, .'UO. 

Logarithmic criteria of oonyergence, 

337. 
Logarithmic series, 197. 
Logarithms, 158, 161-166, 168, 197, 242, 

250. 
Logic, 37, 316, 323, 343. 
Lommel, 375, 393. 
I»ng wave, 383. 
Loom is, 3M. 
Lorenz, 393. 
Loria, XI. 
Loud, 296. 
Lucas de Burgo. See Pacioli. 



Ludolph, 154. 
Ludolph's number, 164. 
Lune, squaring of, 25. 
Liiroth, 358; ref . to, 362. 

MacCnllagh, 311 ; ref. to, 393. 

Macfarlane, 319. 

Machine, arithmetical, 220, 284. 

Maclaurin, 243; ref. to, 236, 244, 280, 
285,290. 

Macmahon, 327. 

Magic squares, 92, 136, 241. 

Magister matheseos, 136. 

Main, 377. 

Mainardi, 334. 

Malfatti, 296, 328. 

Malfatti's problem, 296, 312. 

Mansion, 341. 

Marie, Abbe, 279. 

Marie, C.F. M., 298. 

Marie, M., X., 52, 172. 

Mathieu, 391 ; ref. to, 363, 377, 392, 
398. 

Matrices, 321, 324. 

Matthiessen, X. 

Maudith, 136; ref. to, 14L 

Maupertius, 253, 257, 401. 

Maurolycus, 153 ; ref. to, 156. 

Maxima and minima, 49, 174, 186, 189, 
208, 244, 333, 334, 3:19, 342. 

Maxwell, 387 ; ref. to, 300, 376, 382, 
390, :m, 396, 398, 400, 401, 402, 403. 

Mayer, 398; ref. to, 376. ^ 

BIcClintock, 328. 

McColl, 341. 

McCowan, 383. 

McCuUagh, 311, 393. 

McMahon, 328. 

Blechanics: Greek, 23, 34,43; Stevin 
and Galileo. 158, 182; Descartes, 
Wallis Wren, Hnygens, Newton, 
188, 191, 192, 212-216; Leibniz, 227; 
Bemoullis, 237, 238; Taylor, 243; 
Euler, 253; Lagrange, 2ri6; La- 
place, 274; more recent work, 290, 
328, ^6, 377-381, 401. See Dynam- 
ics, Hydrodynamics, Hydrostatics, 
Graphic statics, Laws of motion. As- 
tronomy, D'Alembert's principle. 

Meissel, 353. 

Meuschmos, 32 ; ref. to, 31, 34, 46, llSb 



416 



INDEX. 



Menelaos, 55 ; ref . to, 57, 157. 

Mercator, 6., 313. 

Mercator, N., 197 ; ref. to, 220. 

M^r^, 182. 

Mersenne, 180, 191. 

Mertens, 336, 367. 

Meteorology, 384-386. 

Method of characteristics, 297. 

Method of exhaustion, 28 ; ref. to, 33, 

36, 42, 169. 
Metius, 154. 
Meunier, 314. 
Meyer, A., 340, 341. 
Meyer, G. F., 334. 
Meyer, O. E., 382, 392, 402. 
M^ziriac, 179; ref. to, 265. 
MichelsoD, 394. 
Middle Ages, 117-137. 
Midorge, 174. 
Minchin, 381. 
Minding, 314. 
Minkowsky, 370. 
Mittag-LefSer, 359. 
Mobius, 294 ; ref. to, 293, 320, 321, 366, 

375, 377. 
Modern Euroi>e, 138 et seq. 
Modular equations, 321), 353. 
Modular functions, 354. 
Mohammed ben Musa Ilovarezmi, 106; 

ref. to, 107, 110, 114, 125, 127. 
Mohr, 300. 
Moigno, 334. 
Moivre, de, 245. 
Mollweide, m\. 
Moments in fluxionary calculus, 205, 

206. 
Monge, 28G-288; ref. to, 248, 259, 282, 

293, .'WO, 314, Ml. 
Montmort, de, 240. 
Montucla, X., 172. 
Moon. See Astronomy. 
Moore, 330. 
Moors, 115, 116, 125. 
Moral expectation, 239. 
Morley, 362. 
Moschopulus, 135. 
Motion, laws of, 183, 188, 213. 
Mouton, 219. 
Muir, XII., 325. 
Miiller, X. 
Miiller, J. See Regiomontanus. 



Multi-coDStancj, 300, 391. 
Multiplication of series, 335, 336. 
Musa ben Sakir, 106. 
Musical proportion, 8. 
Mydorge, 177. 

Nachreiner, 325. 

Nagelbach, 324. 

Napier, J., 162, 163; ref. to, 156, 161, 
164, 165. 

Napier, M., X. 

Napier's rule of circular parts, 166. 

Nasir Eddin, 114. 

Nautical almanac, United States, 376. 

Navier, 387; ref. to, 382, 390. 

Nebular hypothesis, 274. 

Negative quantities, 93, 152, 187, 286, 
372. See Algebra. 

Negative roots. 93, 112, 146, 149, 152, 
166. See Algebra. 

Neil, 190; ref. to, 196. 

Neocleides, 33. 

Neptune, discovery of, 375. 

Nesselmann, 76. 

Netto, 330. 

Neumann, C, 375 ; ref. to, 309, 315, 393. 

Neumann, F. E., 398 ; ref. to, 309, 312, 
390, 392, 396, 401. 

Newcomb, 376 ; ref. to, 306, 307, 392. 

Newton, 201-218; ref. to, 4, 50, 60, 
147, 173, 186, 191, 192, 195, 200, 238, 
243, 244, 252, 254, 257, 258, 262, 268, 
282, 285, 21K), 297, 302, 317, 330, 334, 
372, 380, :«6. 

Newton, controversy with Leibniz, 
227-233. 

Newton's discovery of binomial the- 
orem, 195, 196. 

Newton's discovery of universal gray- 
itation, 213. 

Newton's parallelogram, 217. 

Newton's Principia, 191, 208, 212-215, 

S^£%'f ^«OOy ^TX^^m 

Nicolai, 366. 

Nicole, 240. 

Nicolo of Brescia. See Tartaglia. 

Nicomachus, 72; ref. to, 58, 81. 

Nicomedes, 50. 

Nieuwentyt, 235. 

Nines, casting out the, 106. 

Niven, 396. 



INDEX. 



417 



Nolan, 376. 

Non-Eaclidean geometry, 38, 900-^307. 

Nonius, 153 ; ref. to, 154. 

Notation: in algebra, 15, 75, 93, 133, 
134, 149, 150, 151, 160, 167; Baby- 
lonian numbers, 5-7 ; Egyptian num- 
bers, 13; Greek numbers, 64 ; Arabic 
notation, 3, 73, 87, 102, 112, 127-129, 
159 ; Roman, 78 ; decimal fractions, 
160; trigonometry, 249; differential 
calculus, 205, 221, 222, 260, 269, 283. 
See Exponents, Algebra. 

Nuther, 311, 313, 330, 356. 

Numbers: amicable, 68, 106, 115; ex- 
cessive, 68; heteromecic, 68; per- 
fect, 68; defective, 68; triangular, 
180; definitions of numbers, 372 ; the- 
ory of numbers, 65, 76, 96, 108, 119, 
131, 178-182, 252, 264, 280, 362<372. 

Numbers of Bernoulli, 238. 

Numerals: Egyptian, 13; Babylonian, 
5-7; Greek, 64; Arabic, 87, 102, 103, 
112. See Apices. 

Oberbeck, 386. 

GSnopides, 19; ref. to, 16. 

Ohm, M., 317. 

Ohrtmann, X. 

Olbers, 364, 373. 

Oldenburg, 228. 

Olivier, 300. 

Omega-function, 363. 

Operations, calculus of, 292. 

Oppolzer, 377. 

Optics, 39. 

Oresme, 134 ; ref. to, 160. 

Orontius, 154. 

Oscillation, centre of, 191, 243. 

Ostrogradsky, 333, 379. 

Otho, 142. 

Oughtred, 167 ; ref. to, 147, 161, 202. 

Ovals of Descartes, 187. 

w: values for; Babylonian and He- 
brew, 8; Egyptian, 11; Archime- 
dean, 41; Hindoo, 98; Arabic, 108; 
Ludolph's, 154; Wallis', 194, 195; 
Brouncker's, 197; Fagnano*s, 241; 
Leibniz's, 220; selection of letter w, 
250; proved to be irrational, 259, 
281 ; proved to be transcendental, 2. 



Pacioli, 135; ref. to, 134, 142, 152, 156, 

158, 196. 
Padmanabha, 87. 
Palatine anthology, 73, 120. 
Pappus, 58-61 ; ref. to, 35, 39, 45, 49, 

50, 55, 65, 6(), 153. 178, 186. 
Parabola, 42, 70, 198; semi-cubical, 

190. See Geometry. 
Parabolic geometry. See Non-Euclid- 
ean geometry. 
Parallelogram of forces, 183. 
Parallels, 38, 281, 300, 301, 303, 306. 
Parameter, 48. 
Partial differential equations, 208, 256, 

287, 341 et acq., 379. 
Partition of numbers, 372. 
Pascal, 17&-in; ref. to, 178, 182, 196, 

220, 240, 284, 286, 290, 310. 
Pascal's theorem, 178. 
Peacock, 284 ; ref. to, X., 130, 133, 161, 

283, 315. 
Pearson, 391. 
Peaucellier, 326. 
Peirce, B., 323; ref. to, 291, 817, 376. 

381. 
Peirce, C. 8., 323; ref. to, 37, 307, 321. 
Peletarius, 166. 
Pell, 147, 151, 181, 219. 
Pell's problem, 97, 181. 
Pemberton, 201. 
Pendulum, 191. 
Pepin, 365. 
Perier, Madame. X. 
Periodicity of functions, 349, 360. 
Pemter, J. M., 386. 
Perseus, 50. 

Perspective, 177. See Geometry. 
Perturbations, 273. 
Petersen, 366. 
Pfaff, 341, 342; ref. to, 382. 
Pfaffian problem, 341. 342. 
Pherecydes, 20. 
Philippus. 33. 

Philolaus, 22; ref. to, 28, 68. 
Philonides, 46. 
Physics, mathematical. See Applied 

mathematics. 
Piazzi, 373. 

Picard. E., 347, 360, 360. 
Picard, J., 214, 216. 
Piddington, 384. 



(' 



418 



INDEX. 



Piola,388. 

Pitiscus, 142. 

Plana, 375, 387, 396. 

Planndes, M., 135. 

Plateau, 382. 

Plato, 29-31 ; ref . to, 3, 10, 16, 23, 32, 33, 

34, 35, 63, 68. 
Plato of Tivoli, 109, 125. 
Plato Tiburtinus. See Plato of Tivoli. 
Platonic figures, 39. 
Platonic School, 29-34. 
Playfair, X., 156. 
Plectoidal surface, 60. 
Plucker, 307-309; ref. to, 304, 308, 313. 
Plus and minus, signs for, 150. 
Pohlke, 300. 
Poincarc^ 343; ref. to, XUI., 345, 346, 

347, 352, 359, 368, 384, 400. 
Poinsot, 378; ref. to, 377. 
Poisson, 387 ; ref. to, 175, 298, 330, 333, 

351, 375, 378, 382, 383, 386, 387, 390, 

392, 395, 396. 
Poncelet, 289, 290; ref. to, 178, 288, 

293, 306, 308, 389. 
Poncelet's paradox, 308. 
Porisms, 39. 
Porphyrius, 65. 
Potential, 277, 358, 395. 
Poyntinp, 397, 398. 
Preston, 400. 
Primary factors, Weierstrass* theory 

of, 354, 30)0. 
Prime and ultimate ratios, 198, 212, 

268. 
Prime numbers, 38, 45, 71, 179, 180, 

368. 
Princess Elizabeth, 188. 
Principia (Newton's), 191, 208, 212- 

215, 229, 233, 242. 
Pringfsheim, 336-338. 
Probability. 158, 182, 192, 237, 230, 

240, 245,' 252, 270, 276, 285, 340, Ul. 
Problem of Pappus, 60. 
Problem of three bodies, 253, 256, 377. 
Proclus, 61 ; ref. to, 17, 19, 33, 35, 38, 

30, 50, 54, 58. 
Prop:ressions, first appearance of arith- 
metical and jreometrical, 8. 
Projective geometry, 307. 
Proportion, 17, 22, 23, 26, 33, 36, 38, 

67,68. 



Propositiones ad acnendos iaTenet, 

119,12a 
Prym,355. 

PtolenuBus. See Ptolemy. 
Ptolemaic System, 56. 
Ptolemy, 66^58; ref. to, 7, 9, 54, 55, 

98, 101, 106, 106, 109, 115, ld9. 313. 
Puiseux. 356. 
Pulveriser, 95. 
Purbach, 134 ; ref. to, 140. 
Pythagoras, 19-23, 67-70; ref. to, 3, 

16, 18, 24, 29, 36, 63, 82, 97, 135. 
Pythagorean School, 19-23. 

Quadratic equations, 76, 93, 107, 111, 
112. See Algebra, Equations. 

Quadratic reciprocity, 252, 280, 365. 

Quadratrix, 25, 32, 59, 60. 

Quadrature of the circle. See Circle; 
also see Circle-squarers, w. 

Quadrature of curves, 42, 49, 177, 190, 
193, 220, 222. 

Quaternions, 318, 319; ref. to, 317. 

Quercn, a, 154. 

Qnetelet, 340; ref. to, X 

Raabe, 337. 

Radau, 377. 

Radiometer, 403. 

Rahn, 151. 

Ramus, 153. 

Rankine, 300 ; ref. to, 400. 

Rari-constancy, 390. 

Ratios, 372. 

Rayleigh, Lord, 386; ref. to, 374, 383, 

384. 307, 398. 
Reaction polygons, 300. 
Reciprocal polars, 200. 
Reciprocants, 327, 361. 
Reoorde, 151 ; ref. to, 158. 
Rectification of curves, 160, 177, 190, 

108. See Curves. 
Redfield, 384. 

Reductio ad absurdum, 28. 
Reech, 381. 
Regiomontaniis, 140, 141 ; ref. to, 139, 

140, 153, 154, 155, 158, 160. 
Regula aurea. See Falsa positio. 
Rejnila duorum falsorum, 106. 
Regula falsa, 106. 
Regular solids, 21, 31, 34, 38, 51, 110, 168. 



INDEX. 



419 



Reid, 384. 

ReifT, XI. 

Reuaissance, 13^156. 

Resal, :J77. 

Rt^ye, 2<r0; re f. to, 305. 

Ri'yiiolds, 383. 

Rhaeticus, 141 ; ref. to, 139, 142. 

Rlieticus. ^ee Rliscticus. 

Rhind papyrus, 10-15. 

Ricoati. 241 ; ref. to, 239. 

Richard of Walliiigford, 135. 

Richelot, 353; ref. to, 309, 312. 

Riemann, 357-359; ref. to, 304, 3a'>, 

312, 315, 339, 342, 354, 355, 35G, 'JC'2, 

368, 384, 392, 398. 
Riemann*8 surfaces, 358 ; ref. to, 35G. 
Roberts, 313. 

RoberN-al, 172; ref. to. 172, 1K7, 191. 
Rolle. 241 ; ref. to, 23U. 
Roman mathematics in Occident, 117- 

124. 
Romans, 77-83. 

Romanus, 154 ; ref. to, 142, 148, 154. 
Romer, 199. 
Rosenberger, XIV. 
Rosenhain, 355 ; ref. to, 353. 
Roulette, 171. 

Routh, 380; ref. to, 381, 3!r7. 
Rowland, 382, 394, 397, 308. 
Rudolff, 151. 
Ruffini, 328. 
RUhlmann, 400. 
Rule of signs, 187, 198. 
Role of three, 92, 106. 

Saocberi, 303. 

Sachse, XIII. 

Sacro Bosco. See Halifax. 

Saint- Venant, 390; ref. to, 322, 382, 

389, 393. 
Salmon, XII., 296, 311-313, 330. 
Sand-counter, 65, 90. 
Sarrau, 393. 
Sarrus, 333. 

Saturn *s rings, 192, 376. 
Saurin, 240. 
Savart, 387. 
Scaliger, IM. 
Schellbach, 296. 
Schepp, 362. 
Schering, 306; ref. to, 3S7, 365. 



Schiaparelli. 32. 

Schlafli, 306 ; lef . to, 338, 353. 

Scblegel, 322; ref. to, Xn., 306. 

Schlessinger, 300. 

Schlomilch, 375. 

Schmidt, XIL 

Scbooten, van, 189 ; ref. to, 190, 202. 

Schreiber, 288, 300. 

Schroter, H., 313; ref. to, 296, 353. 

Schriiter, J. H., 374. 

Schubert, 297. 

Schumacher, 366 ; ref. to, 348. 

Schuster, XIII. 

Schwarz, 3(>1 ; ref. to, 297, 330, 345, 

Schwarzian derivatiye, 361. 

Scott, 325. 

Screws, theory of, 378. 

Secants, 142. 

Sectio aurea, 3.3. 

Section, the golden, 33. 

Seeber, 371. 

Segre, 305. 

Seidol, 339. 

Seitz, :m. 

Selling; 371. 

Sellmeyer, 393. 

Semi-convergent series, 336. 

Semi-cubical parabola, 190. 

Semi-invariants, 328. 

Serenus, 55. 

Series, 111, 245. See Infinite series. 
Trigonometric series, Divergent 
series, Absolutt^ly cimvergent series. 
Semi-convergent series, Fourier's 
series. Uniformly convergent series. 

Serret, 313; ref. to, Ml, 342, 377, 37S). 

Servois, 2Ht, 288, 290. 

Sexagesimal HyMtcm, 7, 57, 65, 67, 126. 

Sextant, 2IH. 

Sextus JuliiLs Africanus, 58. 

Siemens, •'M5. 

Sigma-function, 354. 

Signs, nile of, 187, 193. 

Similitude (mechanical), 380. 

Simony, 315. 

Simplicius, 01. 

Simpson, 249. 

Simson, 290; ref. to, 37, 89. 

Sine, 99, 102. 109, 116, 126, 140, 141; 
origin of term, 100. 



420 



INDEX. 



Singular solutions, 226, 265, 277. 
Sluze, 189; ref. to, 222, 224. 
Smith, A., 381. 
Smith, H., 369, 370; ref. to, XIII., 353, 

372. 
Smith, R., 242. 
Sohnke, 353. 
Solid of least resistance [Prin. II., 25], 

215. 
Solitary wave, 383. 
Somofif, 381. 
Sophist School, 23-29. 
Sosigenes, 81. 
Sound, velocity of, 270, 278. See 

Acoustics. 
Speidell, 165. 
Spherical Harmonics, 247. 
Spherical trigonometry, 56, 115, 280, 

294. 
Spheroid Giqnid) , 384. 
Spirals, 42, 60, 237. 
Spitzer, 333. 

Spottiswoode, 325; ref. to, XII., 292. 
Square root, 65, 94, 159. 
Squaring the circle. See Quadrature 

of the circle. 
Stahl, 306. 

Star-polygons, 22, 135, 156. 
Statics, 44, 182. See Mechanics. 
Statistics, 340. 

Staudt, von. See Von Staudt. 
Steele, 381. 
Stefano, 382. 
Steiner. 296, 296; ref. to, 293, 297, 298, 

308, 311, 312, 320, 348, 357. 
Stereometry, 31, 33, 38, 168. 
Stern, 357, 365. 

Stevin, IGO ; ref. to, 134, 162, 182. 
Stevinus. See Stevin. 
Stewart, 2^)0. 

Stifel, 151 ; ref. to, 149, 151. 155, 162. 
Stirling, 244. 
Stokes, 381 ; ref. to, 339, 382, 383, 386, 

388, 390, 393, 398. 
Story, 306. 
Strassmaier, IX. 
Strauch, 334. 

Strings, vibrating, 242, 255, 262. 
Stringham, 306. 

Strutt, J. W., 386. See Rayleigh. 
Struve, 366. 



Sturm, J. C. F., 890; ref. to, 178, 

379, 381. 
Sturm, R., 296. 
Sturm's theorem, 330. 
St. Vincent, Gregory, 190, 197. 
Substitutions, theory of, 292. 
Surfaces, theory of, 250, 287, 296, 299, 

309, 310, 314. 
Suter, X. 
Swedenborg, 274. 

Sylow, 330; ref. to, 380. 

Sylvester, 326; ref. to, Xm., 216, 296, 

310, 311, 312, 319, 324, 325, 328, 390, 
341, 361, 368, 370, 378, 396. 

Sylvester II. (Gerbert), 120-124. 
Symmetric functions, 250, 328, 830. 
Synthetic geometry, 293-^907. 
Synthesis, 30, 31. 

Taber, 324. 

Tabit ben Korra, 106; ref. to, 106. 
Tait, 283, 319, 381, 388, 400. 
Tangents, in geometry, 62, 173, 186; 

in trigonometry, 110, 141, 142. 
Tangents, direct problem of, 196, 223 ; 

inverse problem of, 169, 189, 220, 222, 

223. 
Tannery, 343 ; ref. to, 362. 
Tartaglia, 143-145; ref. to, 152, 15a 
Tautochronous curve, 191. 
Taylor, B., 242; ref. to, 234, 255. 
Taylor's theorem, 243, 268, 269, 333, 

342. 
Tchebycheff, 368. 
Tchirnhausen, 241; ref. to, 224, 226, 

264,328. 
Tentative assumption, 75, 92. See 

Rcgula falsa. « 

Thales, 17, 18; ref. to, 16, 20, 21. 
ThesBtetus, 33 ; ref. to, 35, 36, 70. 
Theodorus, 70 ; ref. to, 29. 
Theodosius, 54; ref. to, 108, 125, 127. 
Theon of Alexandria, 61; ref. to, 37, 

51, 55, 65, 82. 
Theon of Smyrna, 55, 58, 72. 
Theory of equations. See Equations. 
Theory of functions, 268, 269, 344, 346, 

346, 347-362. See Functions. 
Theory of numbers, 55, 76, 95, 108, 119, 

131, 178-182, 252, 264, 280, 362-372. 
Theory of substitutions, 329, 364. 



INDEX. 



421 



Thermodynamics, 385, 398-401. 

Theta-functions, 352, 353, 355, 380. 

Theta-f uchsiaos, 315. 

Theudiiu, 33. 

Thomac, 353, 3G2. 

Thome, 344 ; ref . to, 345. 

Thomson, J., 385. 

Thomson, J. J., 382; ref. to, :«»», 35)7. 

Thomson, Sir William, 3ie, 31K> ; ref. 

to, 283, 315, 358, ,'$81, 382, :w:i, ;W8, 

391, 31»3, 394, 395, 399, 400, 403. See 

Kelvin (Lord). 
Thomson's theorem, 359. 
Three bodies, problem of, 253, 256, 377. 
Thymaridas, 73. 
Tides, 278, 383. 
Timffius of Locri, 29. 
Tisserand, 377. 

Todhunter, 334; ref. to, I., XIII., 375. 
Tonstall, 158. 
Torrlcelli, 171. 
Trajectories, 2d4, 238. 
Triangulum charac^teristicum, 220. 
Trigonometric series, 283, 3:)9, 367. See 

Fourier's series. 
Trigonometry, 51, 50, 98-100, 109, 110, 

115, 135, 140, 141, 154, 1<», 161, 165, 

238, 242, 245, 249, 259 ; spherical, 57, 

115, 280, 294. 
Trisection of angles, 24, 31, 60, 163. 
Trochoid, 171. 
Trouton, 394. 
Trudi, 324. 
Tucker, XIU. 
Twisted Cartesian, 312. 
Tycho Brahe, 110, 139, 168. 

Ubaldo, 183. 

Ultimate multiplier, theory of, 379. 

Ulug BefT, 114. 

Undulatory theory of light, 192, 339, 
379, 392-394. 

Universities of Cologne, Leipzig, Ox- 
ford, Paris, and Prague, 136. 

Valson, XIII. 

Van Cculen. See Ludolph. 
Vandermonde, 278; ref. to, 2fW, 27H. 
Van Schooten, l«i»; ref. to, 190, 202. 
Variation of arbitrary consonants, .'{78. 
Varignon, 240; ref. to, 236. 



Varying action, principle of, 292, 318, 

379. 
Venturi, 52. 

Veronese, 306 ; ref. to, 307. 
Versed sine, 99. 
Vibrating rods, 387. 
Vibrating strings, 242, 256, 262. 
Vicat, 389; ref. to, 390. 
Victorius, 79. 
Vieta, 147; ref. to, 50, 142, 162, 163, 

154, 167, 196, 202, 217, 264. 
Vincent, Gregory St., 190, 197. 
Virtual velocities, 34, 266. 
Viviani, 172. 
Vlacq, 165. 

Voigt, XIIL, 365,393. 
Volaria, 237. 

Von Helmholtz. See Helmholtz. 
Von Staudt, 298, 299; ref. to, 292, 294, 

295. 
Vortex motion, 382. 
Vortex rings, 382. 
Voss, 306; ref. to, 336. 

Waldo. 385. 

Walker, 323. 

Wallis. 192-196; ref. to, 98, 161, IH, 

179, 187, 188, 197, 202, 229. 
Waltershausen, XI. 
Wand, 400. 
Wantzel, 328. 
Warring. 264, 330. 
Watson, J. C, 377. 
Watson, 8., ^1. 

Wave theory. See Undulatory theory. 
Waves, 382-385. 
Weber, H. H., 355. 
Weber, W. E., 394; ref. to, 367, 363, 

388, 396, 398. 
Weierstrass, 359; ref. to, 328, 339, 362, 

353, :i54, 369, 361, 362, 372. 
Weigel, 219. 
WeUer, 341. 
Werner, 153. 
Wertheim, 390. 
Westerpaard, 340. 
Whejitstone, 386. 
Whewell. IX., 43, 253. 
Whiston, 216. 
Whitney, 87. 
Widmaun, 150. 



422 



INDEX. 



Wiener, XI. 

Williams, 267. 

Wilson, 264. 

Wilson's theorem, 264. 

Winds, 384-386. 

Winkler, 392. 

Witch of Agnesi, 260. 

Wittstein, XII. 

Woepcke, 83, 103. 

Wolf, C, 241 ; ref. to, 167. 

Wolf, R., XI. 

Wolstenholme, 341. 

Woodhouse, 334. 

Wren, 177; ref. to, 188, 198, 213, 287. 

Wronski, 324. 



Xenocrates, 29. 
Xylander, 153. 

Young, 392; ref. to, 386, 387. 

Zag, 127. 

Zahn, Xn. 

Zehfuss, 325. 

Zeller, 365. 

Zeno, 27. 

Zenodonis, 51. 

Zero (symbol for), 7, 88; origin of 

term, 129. 
Zeuthen, 313; ref. to, IX., 297. 
Zeuxippus, 40. 
Zolotareff , 371 ; ref. to, 372. 



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